Since there are so many strengths and weaknesses and ways to win, I approach this by asking the question, if I were playing one game blindly (vs AI, I don't really do multiplayer) and had to win, which Civ would I choose? My first list of candidates, from both polls, was Inca, Rome, Greece (Gorgo), Nubia, Sumeria, Aztec. Several other powerful civilizations didn't make the first cut because of dependency on one victory type or starting in the right terrain (Inca really should have been kept off by that consideration except that they're so powerful anyhow, I included them). I've also read much about the virtues of the Maori but have yet to try them myself, so kept them off. In the end it came down to Rome or Sumeria for me and I chose Rome. Early military strength and culture, roads, and all in all great balance.
