Okay, I know the problem. I have just recently updated my windows from Windows ME to Windows XP. Before I had ME, Grim Fandango ran beautifully with the patch. Unfortunately, the problem you're experiencing is an XP problem. Not to worry! Simply download the patch from www.lucasarts.com
... after that, check the update and/or original game readme and learn how to copy the sound and music files directly to your hard drive. The problem will show up again every once and a while, but save frequently and it won't be so much of a hassle. It's worth it to play the game.
If you can't figure out how to copy the music/data/video files to your harddrive after the patch is installed, post back here and I'll give you an in-depth description.