While others were posting about whatever explicitly Asian fantasy bits and bobs there were for 5E (samurai path for Fighters, kensei path for Monks, etc.) I posted my ideas for how to re-fluff the existing PHB classes to fit OA. I'll copy/paste that here for convenience:
Honestly, a lot of the "generic" options are just fine for OA style play. Just looking at the PHB options:
Berserker Barbarian -- works for a Zhang Fei (Three Kingdoms general) style wild and reckless warrior
Totem Warrior Barbarian -- just use more Asian themed animals names instead of the Euro/North American defaults, keep the abilities the same
Valor Bard -- works for a geisha/gisaeng sorceress type
Lore Bard -- wily strategists like Zhuge Liang/Kongming or Sima Yi (again from Three Kingdoms)
Clerics are a bit hard to squeeze in, but a Light or Tempest Cleric could be a Yang-themed magician while a Knowledge or Trickery Cleric could be a Yin magician. War Clerics might be one type of wandering Xia.
Druid -- either type could be an animist shaman: Shinto, Mongolian, etc.
Champion Fighter -- run of the mill samurai or stalwart warriors from other cultures
Battle Master Fighter -- kensei/weapon masters
Eldritch Knight Fighter -- wandering Xia with mystical kung fu
Monk -- the class is already OA themed, any subclass fits
Devotion Paladin -- samurai champions of bushido
Ancients Paladin -- maybe doesn't fit so well
Vengeance Paladin -- a warrior out for revenge against old enemies is a standard plot of many wuxia and manga stories
Hunter Ranger -- also fits really well for a wandering Xia warrior
Beast Master Ranger -- make sure the animal companion is an Asian animal and you're good
Thief Rogue -- a yakuza, ninja, or wandering Xia
Assassin Rogue -- another good choice for a ninja type
Arcane Trickster -- the more mystical ninja, or another wandering Xia type
Draconic Sorcerer -- ancestor is a lung dragon instead of a chromatic/metallic dragon
Wild Magic Sorcerer -- those kooky hermit/immortal magicians with their unreliable magic!
Warlock -- mix and match Patron and Boon to get a Xia warrior type or a crafty Taoist magician type
Wizard -- again any school can work with just a few tweaks to flavor to make a Wu Jen or other OA style magician.
Most of the Backgrounds are generic enough to work as is as well, I'd think.
The idea's been on my mind since then. As others later pointed out, that works for classes, but not really well for races. That's a whole other kettle of fish, but as you could probably guess from both Flying Swordsmen and Chanbara, I'm fine with a 'human only' OA style game. Nothing wrong with the various demi-human races in 1E and 3E OA, but since there aren't any official versions yet, they'd need to be homebrewed which can be tricky/time consuming to get right.
So not only have I been considering how to run a game of OA style fantasy with 5E, I've been considering doing just that. Exactly what I don't need as my players are clamoring for more Chanbara and I've been slow to get that going.
My latest train of thought on the 5E OA topic, though, has been to rework the Backgrounds and leave (as my quoted post above shows) the classes alone. Maybe disallow a few classes or subclasses. But most of the flavor needed could come from slight tweaks/renames of the Backgrounds available. Here's what I'm thinking now (new ideas in blue):
Acolyte: fine as is, change the name if you really want
Chalatan: also fine as is
Criminal: needs two types. Normal criminal (as in the PHB) for people who were arrested and have facial brands/tattoos, and Yakuza for organized gangster/Triad types. Yakuza would get Intimidation and Sleight of Hand for skills. Yakuza would also get an organizational feature similar to the acolyte's, only for their gang.
Entertainer: no mechanical changes, but call it Artist or Geisha or Gisaeng or what have you.
Folk Hero: no changes needed. The name could be changed to Ronin easily, to model all those wandering samurai do-gooders and ne'er-do-wells.
Guild Artisan: no changes mechanically, but make some edits to the list of guilds.
Hermit: no changes needed
Noble: normal nobles are just nobles. Knights (variant) become Samurai.
Outlander: again, not really needed to change this for characters from more barbaric neighboring lands or those who grew up in remote locations. A Foreigner option, for those in the mood to play the gaijin in a strange land would be good. You'd probably want to give two different skills than Animal Handling and Survival, but I'm not sure right now which two would be best.
Sage: Rename this as Scholar. For people who've taken, been studying for, have passed but not taken office, or have failed the Civil Service exams. It's a big deal in Confucian cultures.
Sailor: no real change needed.
Soldier: no real change needed. This should be for rank-and-file troops, mercenaries, or the like though. Officers get their own dealio below.
Urchin: as with many options, no real changes needed. An alternate version for Shinobi would be good, though, replacing Sleight of Hand with either Investigate or Perception.
Civil Official: similar to the Sage but with a special feature more like the Noble. Persuasion and History as skills. Mandarins were the elite class of commoners in China.
Military Official: similar to the Soldier but again with a feature more like the Noble. Athletics and History as skills. This is for military officers, who had to go through an exam process similar to the Civil Officials to gain their posts.