Exotic? Not really the word I'd use, but....
As The Judge has said, Aliette de Bodard wrote the Obsidian and Blood trilogy, which are set in the Aztec empire. She's also written fantasy and science fiction with Vietnamese culture at its core, as well as fiction set in her native France.
Robert Silverberg wrote a version of the legend of Gilgamesh, and one which saw Gilgamesh travel from the Underworld to the modern era, although I cannot remember the name -- it was a while back.
When you say Mesopotamian, perhaps you'd be interested in a more recent setting for fiction in that region? If so, you can do a lot worse than Saladin Ahmed's Throne of the Crescent Moon. There are also some pretty good graphic novels coming out of that region, from what I've been hearing. I'd need to go and ask others about that.
Zen Cho has written Sorcerer to the Crown, which is on my list to buy. She's Malaysian by birth (if I remember correctly) but, in case you're looking for authors from East Asian cultures other than just Chinese or Japanese, I thought I'd add her. Set in Britain, but I believe has characters from East Asian cultural backgrounds (and others) in it.
I've seen JY Yang's work recommended by Kate Elliott; and Ken Liu's The Grace of Kings looks interesting although, from what I gather, it's informed by culture rather than necessarily based on any actual culture of our own world (I could be wrong on that).
Most of the Indian spec-fic writers I know of seem to be science fiction rather than fantasy. Something I should look more into, perhaps.
Finally, if you want a couple of big names producing high quality non-Eurocentric fantasy (and sci-fi) other than from exactly the cultures you specified, then please look up NK Jemisin and Nnedi Okorafor. Okorafor tends to write Afrofuturist fiction. Jemisin is more second world epic fantasy and sometimes sf, but refreshingly not in a pseudo-Mediaeval Europe.
