Thanks for your reply. I got the game to work so I will post what was missing.
I made sure the components you mentioned are installed:
DirectX 9:
winetricks d3dx9_24
Visual C++ 2008:
winetricks vcrun2008
At this point I stil had the problem so I looked at the logs and these errors show when the UI appears:
0080:fixme:d3dcompiler:make_assignment Assignments with writemasks and matrices on lhs are not supported yet.
0080:err:d3dcompiler:compile_shader HLSL shader parsing failed.
Searching the internet for the shader parsing error, I found that you have to install the following:
winetricks d3dcompiler_43
winetricks d3dcompiler_47
After installing these, the game worked. So the main problem was with d3dcompiler.
I hope this thread helps people who face the same problem in the future. It seemed to me that running the game on Linux should not be hard or require a lot of tweaking.
Thanks again for your help.