China
China is probably the oldest semi-continual polity in the world anyone actually gives a shit about. Over the course of twelve major dynasties, a shitload of smaller ones, a bunch of big civil war punch-ups, one Communist dictatorship, and its current, ongoing, post-Communist oligarchy, this long strip of west Asian coast and assorted inlands has had a tremendous, outsized effect on the world economy and the culture of surrounding nations.
Naturally, this has made it fertile fodder for tabletop gaming. From the Forgotten Realms to Golarion, few are the fantasy gaming settings without a "medieval China"-equivalent somewhere in the world. However, quite often, these Sure-Fine brand not!Chinas are about as well-researched and accurate as, well, their European counterparts, taking the broad cultural outline of a big empire ruled by a centralized bureaucracy and an all-powerful Emperor (who may or may not be a god / demigod) and a few specific trappings of architecture and dress to make what amounts to a China-based theme park for the adventurers to roam around in, seeing the sites, taking pictures, and fighting their way through that bestiary full of East-Asian monsters you never get to use.
There's nothing wrong with this, really, but there's nothing particularly interesting about it either beyond the novelty of playing a bunch of slack-jawed tourists in your adventuring campaign.
However, the other major influence China has had on tabletop gaming is through the medium of wuxia.
Wuxia
"'Wu' means martial arts, which signifies action, 'Xia' conveys chivalry. Wuxia. Say it gently... 'whooshah'... and it's like a breath of serenity embracing you. Say it with force, 'WuSHA!', and you can feel its power."
— Samuel L. Jackson, "The Art of Action: Martial Arts in the Movies"
Thank you, Reverend Jackson.
Wuxia is what China has instead of Tolkien. Just as the Western fantasy setting has got your dwarves and your elves and your dark lords leading armies to conquer the world, China has Jianghu, the Land of Rivers and Lakes, where corrupt civil authority forces noble wandering heroes to live like outlaws as they fight to restore order, learn secret techniques from old masters, are forced to battle their former best friends, etc. Just like Western fantasy, there's a lot of high-brow, literary stuff, but there's also a lot of entertaining trash pumped out to fill a public need for it. For instance, those cheap Shaw Bros. kung fu movies are wuxia, but so is Once Upon A Time In China.
And, naturally, this genre has its own tabletop games.
The biggest success is probably Exalted, White Wolf's epic fantasy role-playing game. While there are, obviously, a shitload of other influences, from a corrupt cosmic bureaucracy and physical Realm in need of heroes to fix things to the super-martial arts and flowery naming conventions, Creation would simply not be recognizable without the trappings of wuxia. This is true even in a subtler sense: wuxia often focuses on tragedy and deeply-flawed heroes whose best intentions turn on them. Thanks to the Great Curse, all the exalts are, unless they do their utmost to defy their fates, doomed by the stars.
Other games, like Legends of the Wulin and Feng Shui draw on the genre more overtly. Even if the latter is more about aping the whole spectrum of Hong Kong cinema than wuxia specifically, even the later "heroic bloodshed" films are basically wuxia pictures set in the modern day with guns instead of swords, cities instead of forests, and cops and triads instead of heroes and bandits.