After giving it some tought I came up with my guess for the Mayans, something to reflect their focus on calendars and how they used it to develop their intensive agriculture.
UA- Land of Pheasant and Deer
Gets a Free great Prophet with calendar, recieve faith every time a city grows.
UU: Atl atl, I guess its a composite bowman with bonus movement.
UI: Cenote Shrine- it replaces the Lumber Mill, +2 hammer, (another +1 after reseaching Civil Service), counts as fresh water source. Can only be built on jungle tiles.
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Ok the idea behind this is to have the Mayans being able to create a tall religous empire with a strong middle game.
Playing Mayans you should strive to making your cities grow, even if the faith bonus was small, it would help in spreading your religion. Free Prophet at calendar helps in this regard, plus you want to get those plantations as fast as possible anyway.
The jungle is a production hell...not for the Mayans, here's where the UI gets into play and its very important. The Mayan lumber Mill replacement, the Cenote Shrine, generates +2 hammers (to override the jungle penalty) and allows you to get that needed extra production. It also works as Oasis, allowing you to generate enough food to keep your cities growing. Spawning without rivers and hills in the middle of jungle oceans doesnt sound so bad now..wich is precisely what the Mayans did in real life.
Late game The mayans are not as competitive due to their faith focus, but can still manage, get University and your jungle tiles get even more valuable. Even if you dont spawn near jungles, the bonus faith on growth and free Prophet would be helpful.
So, what do you think?