While only a few of the songs from Civilization III have names, many of them still stand out, like the Ancient Era Theme for the... well... Ancient Era, and especially the Modern Era songs: Smash, Techno Mix, and Stars Full.
Civilization IV: Baba Yetu, which has become a Leitmotif for the entire series. In fact, it has won a Grammy, the first ever piece of computer game music.
While not part of the main game, Mado Kara Mieru, also by Christopher Tin, is used as the main theme for Realism Invictus, a long-running Game Mod of Civ 4.
Most of the background music in Civ IV is this way. It's been described as "one of the few games that lets you change the background music, and one of the few games where you don't need to".
Just the fact that every leader got its own piece, which is performed by increasingly more (modern) instruments as the civilization advances. And especially the tunes that are newer versions of old themes, which give an awesome moment of nostalgic realization, and of course, an opportunity to remember how amazing those songs were as well.