Sounds like a video card problem. Sometimes old games don't like new video cards...You should
- Try to patch the game
- Update your video card driver
- Play a little with your video settings (resolution, refresh rate, etc)
- Try to run the game in Win98/Win95 compatible mode (right click on the Diablo shortcut, Compatibiliy, Win98/Win95)
On the Blizzard support page, I have only found
this
Edit: maybe
this thread can help you too