I'll just separate them into their genres for easy access...
First-Person Shooter
Sauerbraten - Cube dazzled me with its level editing prowless, so I played Sauerbraten, essentially Cube 2, and I was amazed. Furthermore, this is the must-have game for level editing, but its online play is severely hindered by hackers and know-it-all newb haters.
Tremulus - This is one of those proof-of-concept games. They thought to combine two totally different gameplay aspects into one game. The two genres this can fall under are FPS and RTS. This game is a bit like Team Fortress in the sense that you can choose a character class, and it has a familiar building feature to build turrets and similar objects. I really like this game. Click here to learn more about it.
Warsow - I just like the graphical style of this game. The actual gameplay isn't impressive, but it is still entertaining if you like first-person shooting. My advice is to find something else, though. Also, the only way to get the Linux version of the game is to get the full package, which includes the Windows binary.
Platformer (you know, the jump-and-collect stuff)N - This was made in Flash, but there's a Linux binary available for those who're still anti-Flash. It's one of the best platform-jumping games I know of, and I hope you enjoy it. This game has a level editor.
Rhythm (like Guitar Hero)Frets On Fire - I shouldn't be the voice for the rhythm genre, because I'm not a big fan of that stuff, but I loved PaRappa the Rapper and its sequels, and Frets is not like PaRappa the Rapper.
Strategy (anything that requires thinking)FreeCol - The screenshot in this blog post is of this game. FreeCol is a pretty neat game of colonization. Your mission is to colonize territories. This game is far from done at a development standpoint, but there's enough to keep you busy for a while.
Gate88 - This is one weird game. By weird, I just mean unorthadox, and, by unorthadox, I mean weird. Anyway, I like this game, and I think you will, too. It has a blend of action-packed and strategic gameplay, and it looks good to boot.
Glest - This is my favorite strategic battle game. I don't play it often, though, because I can't save my progress and feel the satisfaction of leveling up like in Diablo or some other dungeon-crawling game.
UFO:AI - The AI is for Alien Invasion; not Artificial Intelligence. However, it's a futuristic strategy game that doesn't give you the build-to-do gameplay mechanic. Instead, you just send in troops to kill aliens.
Wesnoth - Rather "the Battle for Wesnoth," as the site says, this is one of the games I ended up playing on Windows first. I like it, but it's not quite as addictive as
Glest.
Graphic Novels (not-so-interactive stories)Narcissu - This game is like reading doujinshi (Japanese comics) but requires electricity. The payoff is that it's free. There's a sequel as well, so try
Narcissu 2 if you like
number 1.
Obviously, you can play Flash-powered games in Linux, too, so I suggest a game called
N.