What Operating System are you using? If it's Windows 2000, then it can be a little hit and miss when it comes to loading Grim Fandango. Try updating to the latest Service Pack by using Windows Update.
Also, right click on desktop and select properties, and lower the colour to 16-bit, that may help.
If you're using XP the following doesn't apply (because you can't do it):
Uninstall DirectX - See this thread
- and then reinstall it using the version you have on the disc. You'll need to run Grim straight from the *.exe file in it's root directory now (C:\Program Files\LucasArts\Grim), as it may refuse to run it from the Pop-up dialogue.