Yes you can, but is it easy to do for someone new to Linux? No it's not. If you are just planning on playing games I would stick with Windows, that's not what Linux is around here for.
DirectX is made and run by Microsoft, so Linux must emulate it's enviornment in some way. This is done via Cedega (if you want to pay a monthly subscription) and Wine (which I hear is VERY difficult and has alot of flaws on top of not supporting that much in the first place).
However, any game that uses OpenGL can run native to Linux. Meaning you just have to install it via command prompt like anything else. These games include Unreal Tournament, Doom, Americas Army etc.