Well, the AIs ranking high in this poll are militaristic leaders, and choosing better units to build would make them do better.
Mongolia, the Aztecs and Denmark are in a league of their own in terms of being uncompetitive. If they are going to keep their character, they all need better military performance, as Thal said. Mongolia and the Aztecs also scored in the top three in terms of being played the least in the other poll. So with the latter two, the goal is to make them both stronger for the AI and more interesting for the human.
Denmark: like Mongolia, they can't do much with any of its traits. Historically Berserkers were called that because they fought as if they weren't hurt, much like the Japanese UA. But could we give all their units March? Obviously, the UU could also be made even stronger. (And of course the Norwegian Ski Infantry is much easier for the AI to use.)
Aztecs: their UA would gain strength, and their bad attitude would pay off more, if the Jaguar Warrior could do better and last longer. Buff it +2 and the Aztecs become a genuine threat to anyone around them, as well as more fun for the human racking up culture (making them even more flexible). They could also pick up MortalD's suggested woods/jungle healing. (This may seem like a lot, but keep in mind that it shouldn't make them more OP early on than Japan or, in a different way, Persia and Greece.)
Mongols: the downside for humans is that, like Denmark, all they do is warmonger - and they do it one way only (Keshiks with the Khan). Knowing it's Thal's favorite civ, I'm not going to suggest we vary them the way other 2-UU civs may be. My suggestion is that the Khan either be removed or replace the UA, then add a Horseman UU that has March (no Khan) or +1 movement, or both.