I used to play it a decent amount. I've mainly picked a bunch of different Civs.
I like the Vikings (I think Denmark is Civ) since I have Norwegian blood in me and I consider myself a Viking first before anything else (including being an American).
France was alright because I think they get a nice culture boost for the first half of the game before the tech that disables it. I was using them in a cultural victory but I haven't finished that game yet. I actually might have made it a little hard to win considering I started conquering Civs which makes it harder to build up the culture points easy (since the more cities you have the more it requires). I'll just have to go for the main Civ that is a little ahead of me.
I played as the Greeks and Egyptians also. I loved the Egyptians since they build wonders faster (I believe that is their perk). I did pretty good with both races.
I liked Civ 3 the most since the users at CivFanatics supported it so much with awesome user made scenarios. I love Civ but the AI is dumb as shit most of the times. Also, if you turn the difficulty up it just has the computer start with unfair advantages (more troops, etc). It doesn't make them smarter just has them "cheat" more (such as trading techs with each other a lot and screwing you over).