The best case was if the game had a native Linux version, which most do not have. Your second obvious choice is to run it through Wine, in which case you'll want to check the AppDB entry for that game
to see how (different versions of) the game have worked and if somebody has tweaks for it. Wine isn't just for gaming, however, and you may be able to get better results by using one of the things that are actually made with gaming in mind, such as Cedega/TransGaming
. They're typically non-free, e.g. you have to pay something to use them, but then again, you would have to do that for a legal copy of Windows too (or a game console), right?
DirectX is a Windows thing, it's one way to do things but it's not the only one. If a game is designed such that it only performs well on Windows (e.g. it's near impossible to achieve the same performance under other systems, or requires an insanely big amount of work to get there), then you really should play that game on Windows if you want to play it. At this point I recommend getting a game console, because most games are available on those and with less trouble (drivers, OS version, installation trouble, ...)