Today, I installed TrackMania United Forever, and experienced the same problem, when trying to launch the game.
Therefore, I decided to try reinstalling Windows 8, and now it actually works right out of the box!
I think the problem was, that the first time I upgraded to Windows 8: I chose to keep my settings, files, software, etc. from Windows 7.
That's probably not the best thing to do.
When I'm looking back, I think the exact problem was, that I ran some gaming tracking software, that uses DirectX, and if I've chosen to keep the software from Windows 7, then I've probably also decided to keep the DirectX settings from the old version of Windows.
Now everything works fine, both the tracking software, and the games themselves.
But, thanks for your suggestions anyway!