The Aztecs and Maya do represent a good amount of ancestors whether they may be distant or close for Guatemalans (Which I am half Kaqchikel so have some representation here), Mexicans, etc.
But like others have said sounds like you want a more Caribbean based civ. I believe someone made a mod for the Carib or Arawak (Can not remember which one) for civ 5 which is entirely playable if you look for it the mod download screen in game. It comes in an "Americas Civs" mod which includes other civs like the Inca, Zapotec, Chibchas, Tupi, etc.
So there is a mod at least if you want to try them out.