I play random on MP now, before I easily made it into the top 5 just by playing as China and Romans. I got bored playing the same strats over and over. I must say, I've played against the Aztecs more than any other civ, and have only lost maybe twice agaisnt them (they got a horseman army extremely fast on both occasions).
There are defiinatlly civs that are overused, and some that are easier to play, but now having played all civs at least 5 times on MP and probably 10 times that amount with each civ on SP, I've gotta say it's way more satisfying to beat someone with a weaker civ or what other's perceive as weak. Today I out teched and totally dominated a Japanese player using the french, and he was teching at a pretty fast clip. I just got a jump on him due to finding some other capitals and some artifacts.
Usually if you're playing someone that is using one of the overplayed civs, they usually have a pretty narrow minded inflexible strat that if foiled, they'll get frustrated and make a lot of mistakes trying to adapt. Another benefit of having a preceived weaker civ is that they'll underestimate you, and when you start smashing on them, they aren't prepared.
The civs I fear the most:
Arabs
Americans
China
English
Everyone else I've played agaisnt way too much to not know how to counter their most likely strats. This civs can pack a huge military punch at any point in the game.