Oral lore among those who play has it that the Hispano-American game of Dudo (Sp., lit. "doubt" or "I doubt [it]", but also the word used to call a bluff made by the previous player) is the oldest of the games in the family collectively known as "liar's dice." The game is also called Perudo, after its supposed origins in Peru. (It's certainly known there, as well as in Spain.) The story goes that the game was brought back to Spain by the conquistadores, and all other variants flow from there; many people say specifically that the rules were taught to Francisco Pizarro by his prisoner, the Sapa Inca Atahuallpa.
There's no evidence for this origin, and plenty of reason to find it highly improbable. But the traditional game family is well worth playing, and the diversity of customs surrounding its play is worth a look.
The first reason to doubt the story is that we know dudo not to have been a native South American dice game. It's simply impossible. There were no native South American dice games prior to the arrival of the Spanish, as the new world was wholly lacking in dice. Dice date back thousands of years in the old world, as supported by archaeological finds; we also have ample written documentation of sheep knucklebones being used in a dice-like manner by the Romans. (It's commonly said that we have it for the Greeks as well. This is a mistake; the Greeks also used knucklebones for gaming purposes, but they served as jacks rather than as dice. This use continued for some time thereafter, not dying out entirely until modern manufactured jacks became cheaper and more available than bones.) In the new world, on the other hand, no dice have been found which predate European arrival, nor have knucklebones been regularly found separated from the rest of the animal as would be expected if they were being used as an alternative. It seems, therefore, that there were simply no dice in the Americas, natural or man-made, until Europeans brought them there.
Nonetheless, it's within the realm of possibility that Spanish soldiers brought dice to Peru with them (in fact, it's somewhat likely), and that these might have been picked up by natives who rapidly innovated a game based on their tastes (tastes derived, presumably, from non-dice games that would have had rules entirely unlike the highly randomizer-dependent game of dudo). It's then conceivable that such a folk game would have worked its way up to the point where the Sapa Inca himself could have become a devotee by the time of his imprisonment, and then taught it to Pizarro. It seems a highly improbable chain of events, but not an impossible one.
For such an unlikely occurrence, however, we should like to see a bit more evidence than modern tradition among players ascribing an early origin to their pastime. We should expect to see evidence that the origin story was known sometime prior to the modern era, that the game itself was played within a century or so of its alleged introduction, and ideally that there is some property it shares with native games that is absent from European ones. After all, these things would all have to have been the case for the story to be true and to have persisted into present oral lore; we in fact have no evidence for any of them.
The earliest sources for the story about Atahuallpa teaching Pizarro to play dudo that I have personally come across are from the 1970s, though references to it being a game of Peruvian origin are older. In fact they're nearly as old as the first mentions I've found of the game itself, which are from the late 19th century. I acknowledge the existence of earlier mentions, however, as I've found precisely one citation of an 18th century source which I have been unable to track down myself. Therefore, I presently believe the game to have originated in the 18th century, but without examining the early source I can't be sure it isn't a misunderstood reference to some other game featuring dice and cups, such as an early form of Yahtzee. (Yahtzee and liar's dice could even have a shared ancestor, as they both involve players rolling their dice under a cup and hoping for them to combine in a favorable way. It's not a hypothesis I have any evidence for or against, as I know nothing at all about the birth of Yahtzee; it's something worthy of future research.) I think it likely that the game originated in Peru, and spread with some speed to Europe. This would have the entire family of such games being about two and a half centuries too young to have been played by Atahuallpa.
(I should note a limitation of my research, which has involved using Google Books to try to identify references worth chasing down: finding Spanish sources is nearly impossible. That's not because they aren't archived, but because the Spanish name is a common word with an unrelated meaning, whereas the English phrase "liar's dice" pretty much has to be a reference to the game family we know today. Since the game likely originated in the Spanish-speaking world, this is a serious drawback. The variant name perudo, which is unique to the game, evidently does not have a very long pedigree, but this is not a reason to doubt the age of the game - especially as that name has more currency in English than in Spanish.)
I know of three major versions of liar's dice, each with many small rules variations available, and each with its own jargon that varies a little from group to group and makes the game more interesting to the folklorist. The first, sometimes called "common hand" liar's dice, is exemplified by dudo, and includes almost all commercial versions I am aware of on the market. (Commercial liar's dice sets tend to only include enough equipment for a four-player game; I find this to be almost minimal for what is interesting.)
In common hand versions of liar's dice, each player has a cup and some number of dice. All players roll their dice under their cup, and look only at their own dice, but then attempt to make statements about the set of all dice under all cups. Play proceeds about the table with each person saying something like "three threes" (or "three treys;" in groups that play in Spanish, there's a name of that sort for each of the six faces of the dice). The next player can either make a different, higher bid, or say "I doubt it" or some such phrase to call for the answer to be verified.
Some restrictions are placed on what calls may be made. In the version I'm most accustomed to, neither the value of the face nor the number of dice showing the indicated face may be reduced from the previous bet, and one of the two must be increased. Ones are wild, counting as whatever value is currently being counted, and bluffs are called by saying either "doubt" or "dudo." Of the player who made the last bid and the one who doubted it, the one who was incorrect then gives up a die and starts the next round; obviously with a smaller fraction of the total dice in play being within their ability to look at them, this player is at a disadvantage. A player who is down to one die is said to be prolifico (lit. "prolific"), and starts all rounds regardless of who has lost dice most recently. (If two players are prolifico, the one who has been so for longer has this privilege.) A player who loses all their dice is eliminated from the game.
Some versions of liar's dice involve players making assertions only about what is under their own cup. Most commonly, these involve but a single cup which is passed from player to player, though I've heard of exceptions. In one version, a pair of dice is used; different combinations which appear are of higher and lower rank than others, and a player receiving the cup may either call the previous player's bluff (winning the call if the roll is lower than stated and losing if the roll is the same or higher) or, without looking at the actual answer, reroll the dice, peek at them, and declare a roll higher than the one previously mentioned. Many specific rolls have names; for example, a roll of a two and a one is in many play groups known as a "Mexican" and is one of the highest-ranked throws in the game. This version is commonly played as a drinking game, with the loser of any given challenge having to take a drink.
Another version, which I find highly enjoyable but at which I am terrible, uses five dice under a single cup; often, the dice are labelled A-K-Q-J-10-9 instead of 1-2-3-4-5. The bids available are based on poker hands (leaving out the flush for obvious reasons). On receiving the dice without calling a bluff, the player looks at what they currently show and rerolls any number of them they choose before making their own claim about the hand they now have; the number of dice rerolled is not necessarily disclosed to the other players. Again, the bid must be a higher-ranking one than the previous bid; again, a hand better than the one bid is accepted.
In the poker variant, a wrinkle is added that bids may be ambiguous, and mean the lowest currently-legal bid that fits the declaration. Thus, a player can bid "pair of jacks" to start the game, but they may also say "pair," meaning only a pair of nines. A follow-up bid of "two pair" means tens and nines. This permission of ambiguity allows the bid of "better," meaning only any hand better than the previously bid one; after a sequence of several consecutive "better"s it can be very easy (especially for inexperienced or intoxicated players) to lose track of what, exactly, is being promised. For many players, this confusion seems to be part of the object, or at least seen as being of tactical value. I am told that some players also deliberately count duplicate faces in an idiosyncratic manner wherein three of a kind is called "three pairs," a full house "four pairs," four of a kind "six pairs," and five of a kind "ten pairs."
Obviously, the poker variant of the game postdates the invention and spread of poker, and so must be older than the probable eighteenth-century origins of the family; the two-die variant seems like a stripped-down version of the poker-like game, but the evolution could easily have gone in the opposite direction. I therefore suspect that the common ancestor was either a common hand game or at least a game in which each player has their own cup and dice; it's also highly probable that common-hand liar's dice at the very least has its origins in Spain or South America. Precisely how "this game comes from Peru" morphed into "this game comes from Peru in the early sixteenth century" to "this game was introduced to Spain by Pizarro himself, and he got it from Atahuallpa" is unclear, but it's the same kind of myth-making that leads many traditional observances to be commonly presented as much older than they actually are. It's a claim I've heard parroted without verification by early modern reenactors, who should be warned away from it because the game, delightful as it is, does not seem to be within our period of study. It's also very probably the claim that led to a common-hand variant being played onscreen in the Pirates of the Caribbean films; in fact, these stories are clearly meant to take place prior to the Port Royal earthquake of 1692, which puts them at least a generation too early for the true origins of the game. And, for the sake of intellectual honesty, it's not a claim which ought to continue to be transmitted when teaching the game to others; as appealing a story as it might be, the evidence does not support it in the least.