Babylon, Korea, and Venice are all great, but they're soooo boring... 5> cities, tall tradition, hunker down build the science up or get those trade routes up and going, build a few ships to protect them, buy all your buildings, wonder spam, bribe every city-state blah blah they're soooo good but soooo boring.
Poland is obviously great because of the 7 free SP's, as long as you don't get steamrolled by an early AI. They're versatile, they can go for any victory type. The Ducal Stable is a nice UB that will net you a few extra GPT. And Winged Hussars are a nice UU that buffs the Lancer and make them less useless (does that even makes sense?). They're nice once you can finally get them to Helicopter Gunships.
Arabia are great too, especially on Pangaea. Their UA is better since Caravans will likely be the better form of trading, so you'll have a leg-up there. You can also spam Camel Archers and go on a murder spree, and the Bazaar will keep your economy upright and happiness in check as long as you can keep the AI's friendly enough to trade with you for reasonable prices.
Maya and Siam are pretty good as well, they benefit from great UA's. When I play Siam, I love going left side of Liberty + Tradition, researching Education, and getting 2-3 free Universities with a passive culture boost too.
Shoshone and Persia are probably the best pure Liberty civs in the game, Shoshone get additional land that can work optimal tiles immediately without having to wait or spend gold. The superscouts are nice, can get city growth, culture, and techs from ruins without having to go through anything else. You can eventually upgrade your superscouts and get archers that ignore terrain cost. Comanche Riders are nice rush units that you can pump out pretty quick. Plus you get a bonus across all units when it comes to defending your lands.
Persia is nice too, Liberty will give you more luxuries and more happiness buildings to try and trigger golden ages quicker while your cities are small, plus you will get a free golden age from getting the policy in Liberty, and could also potentially get one with the finisher although a GE or GS will likely be more useful. Immortals are nice, double healing is a pretty effective promotion that will carry over for later units if you aren't using them too much. Satrap's Court is nice for the extra happiness when your cities start to grow, and will grant you more money, the happiness will help you trigger more extra length Golden Ages. Chichen Itza stacks nice with their UA.
All those civs are pretty effective and versatile if played correctly. In my opinion.