Sooooo... this is awkward. SOME of you can now enjoy Batman: Arkham City on your PC as it was meant to be played, but not everyone. The game has been having some issues with its DirectX 11 support, and a newly released patch attempt to address that. You'll need to be running a 64-bit operating system though, in either Windows Vista or Windows 7.
The patch from Rocksteady Studios is detailed on Steam's website, with the added note that those running Windows Vista or Windows 7 in 32-bit should continue to stick with DirectX 9. There are other fixes in the patch as well, but DirectX 11 support has been an issue for the game since launch.
If you don't know which OS type you're running, open up the Windows Control Panel and click the "System" icon. You should see your system type listed right in the middle of the page.