I rarely have an issue with Russia in my games for some reason.
Rome is always a runaway, it seems. Augustus is just nice enough to everyone to let him get a little edge, then all the sudden he hates the world and goes Rambo on everyone. Persia tends to do well too, but I don't know why, as sometimes he's the runaway on the other continent, and sometimes he has a tiny core of cities and a million CS allies.
The two biggest threats though are India and Brazil. Gandhi and Pedro both play the diplo game to perfection, both run a big enough army in general not to get crushed (note the in general though, both can get beaten down early), and both will make your life a living hell if they can survive into mid game.
India is more likely to get there, but less likely to be a major issue. Gandhi always plays puppet master, though. In a tight race one game, he was a runaway (as in early a civ lost it's capital, and Atilla's Court was in Indian lands), and he had something like 20 uranium in his boarders on the other side of the world. Gandhi canNOT have a ton of uranium. Ok, I went there, ready to strike fast and take a few cities. On a huge map (I don't think anyone other than the Huns were eliminated), the turn after my DOW, every...single...civ...declared on me! More than once have I looked at Gandhi wrong and gotten chain denounced.
Brazil still plays puppet master, but is less likely to nuke you back to the stone age late. However, he can actually...you know...win. Like India, Brazil can be hard to crack without losing all your friends, but Brazil sits on its tourism, get a golden age, and all of the sudden they're threatening culture. Pedro loves his CS's too, again, making it hard to keep allies without risking a chain denouncement.
It seems like they both tend to do well because no one can really cross them.