My current favorite is Eador: Genesis. As Psyringe alluded to, there's no real down time in the turn based format, per se. It's just that you can do things at whatever pace you desire (for the most part, anyway). I can take as much time as I want to plan moves, but it's not really the game imposing the restriction on me. In Eador, I click the 'End Turn' button, the game cycles for a couple of seconds and then it's back to the game.
Why do I enjoy Eador: Genesis specifically? For one, I enjoy fantasy-themed turn based games in general so I've gone in with a bit of a positive bias. I like the look of the game. The graphics aren't breath-taking, but the color pallette (sp?) is pleasing to the eye...vivid...nice contract. I like the depth of the game's strategic elements. You have to balance combat, economic, and diplomatic considerations pretty carefully. There's no building a killer stack from the get go and steam-rolling through the map. You need to not only take immediate needs into account, but think several turns in advance, as well. As with most good games in the genre, there's no real 'right' choices, just what you think is the best available option at the time. What puts it over the top for me is the replayability factor. Not only does it possess the 'just one more turn' factor (for me, anyway. But I can't wait to start another game and try a different strategy even if my last game ended in a horrible flame out.
Hope that helps.