I just tested and Crysis works nicely under Wine. I installed it via PlayOnLInux (Wine 1.9.15 32bit) using the option to install a non-listed program.

If you encounter any issues, WineHQ has many tips to get it running.

The first time you start the game, you need to use emulated desktop from wine's settings.

Good! How about Warhead then?

I just test it using the 64 bit version. It works also. Does not forget to install d3dx11 if you use PlayOnLinux (I have sound, but no image without this component).
I don't have Warhead, cannot test.

Yes. That works too.

I never needed to use the emulated desktop.

I seem to get a little better performance in Windows though. Not in terms of average framerate, but there's less random stutter.

I hate POL along with winetricks. I always prefer to do things manually.

Warhead works, but thinks it's been pirated. After a while, all the guns will start shooting chickens.

Good thing they are working on fixing that.