Apparently I haven't had time to watch any videos in quite a while (since I upgraded to Tiger). I just went to watch an .avi in Quicktime 7 and received a dialog box with the message "Some necessary QuickTime software is missing. It may be available on the QuickTime Web site..." It's obviously missing the codec which of course is not available on the QuickTime website.
I know I went thru this excercise before but I couldn't remember the details. You can still open the file in QuickTime, even though it won't play. Then go to "Window > Show Movie Info" and look at the format to see what codec is needed. In the case of pretty much all of my files it's the XviD codec, which requires DivX. The DivX website states that DivX 5 does not support Mac OS X 10.4 or QuickTime 7. After some googling I found a beta version with support for Tiger/Quicktime 7. Once you've got that you can get XviD. After installing both of those, all of my video files work again.