After you installed any of the games you may get a performance boost if you use the latest directX 9 version.

The easiest way to do so, is using winetricks:

winetricks directx9

Read more about winetricks, here:

http://wiki.winehq.org/winetricks

There are some reports that DoW crashes in some situations when using specific version of ATI's fglrx driver.

It seems to be clear that fglrx version 9.3 and 10.12 are affected, while 10.10 seems to work fine.

 Crashes occur in (at least) these situations:

• DoW and WA: When a scout dies.

• Building a Tau Commander.

 There are several possbile work arounds:

• Use another driver (10.10 seems to work) or try your luck with the free radeon driver

• Disable glsl in wine's registry:

- Go to the console
- enter the following command: wine regedit
- The registry editor will open
- Go to HKEY_CURRENT_USER
- Go to Software
- Go to Wine
- Go to Direct3D or create it,  if it does not exist: right click "Wine" and
create a new key named Direct3D
- Mark the new Folder(Key) Direct 3D, add a new "String Value" (Zeichenfolge
for german users) named UseGLSL
- DoubleClick UseGLSL and enter "disabled" at the content-field
 
Thanks to jost for the detailed information.
 If you are affected, you may want to have a look at the bug report as well:
http://bugs.winehq.org/show_bug.cgi?id=19232

If you have problems with completely corrupted textures the problem might be that your systems lacks a library that is used for texture compression in Dawn of War.

Under Debian based system install the package "libtxc-dxtn-s2tc0" and try again.