I tend to spring up and recommend a few free games in these kinds of threads (although I'm not an activist, but heck, these ones ROCK).
Battle for Wesnoth is a lot more combat-oriented than Civilization, but since I like both of these I assume they do attract same types of people. It's a turn-based tactics/strategy game with a high fantasy setting, hexagonal tiles, plenty of inbuilt campaigns and modding capabilities. I really like the gameplay, and the art is really swell. The one thing I don't like is the amount of fantasy clichés - the campaigns are enjoyable anyway, but I've had some facepalm moments with some of the most recycled, dramatic and foreseeable stuff.
Warzone2100 is a RTS that has a lot in common with Command and Conquer games, but has wider tactical elements so you may find it appealing if you like Total War. Unlike many other RTS games, Warzone 2100 has very powerful artillery. However, they need spotters - it is possible to cripple entire batteries simply by destroying their eyes. Nice! The game's not as pretty as Wesnoth though.
And then some good oldies. You may find these at Abandonia (but that, I think, is illegal for most games)
Sword of the Samurai is a weird mixture of RPG and strategy. You step in the shoes of a Japanese samurai with a small army and a small estate. Your superior, a Hatamoto, will give you quests (which are solo missions and award Honor) and campaigns (which are army combat missions and award Land), which you and your three competitors may try to fulfill. Your goal? To become the next Hatamoto by being his best subordinate when he dies. How you do it is not important - you may just murder the other three and outshine their relatively inexperienced heirs, work hard by kidnapping enemy clansmen and doing quests or just stick home and train. When you're the Hatamoto, you'll answer only to your Daimio, who will hand out much harder tasks. The drill is the same otherwise. But when you seize the throne, the game changes and becomes a game of conquest - you only win when the rest of Japan recognizes you as their Shogun!
Mercahnt of Venice (rerelased under the name Machiavelli: The Prince) resembles Civilization a lot. Instead of Civ, though, you play as a Renaissance merchant with three competitors (video game industry seems to like 4p Free For All) and your quest is to be the richest of the lot. You do it by buing low and selling high, of course, but there's more to that. Hire cardinals to collect indulgences for you, and you may even get a hold on Papacy! Bribe Senators, get a Government post and the funds associated! Hire mercenary armies and beat the [poo] out of those smirky Florentians! Have assassins kill cardinals and Senators (or even the Doge or the Pope) controlled by enemy families! Commission valuable art! Renaissance is fun indeed! (note: This game has at least one sequel, but it's pretty bad according to what I've heard) I wholeheartedly recommend this game.