On the other hand, their UU sucks, so ...
Eh? What's wrong with cheap archers?
The Mayans are very powerful, but they do have a hidden flaw - their Unique Ability (gaining free great people, but never of the same type twice) increases the cost of gaining other great people.
So say you're going for a science victory - the Mayan Unique Building gives extra science, so it makes sense. You pick a Great Scientist as one of your Long Count great people, all well and good. You then have to pick Artists, Engineers, Generals, Admirals, Merchants, and every time you do, the cost to generate another Great Scientist by normal means increases.
You can say the same thing if you're going for a cultural victory - you get one Great Artist for free, but the others are more expensive than normal.
I think the Mayans are generally strong, but there are better choices for every victory type. If you want domination, pick someone like Japan, or Germany, or even somewhere like Arabia that has an overpowered horse archer type unit. If you want Cultural victory, places like India or Egypt do those better than the Mayans. If you want science, there's Babylon or Korea. (though I will admit those are both DLC civs)