OK. I guess ReminderFox would work if I set Firefox to come on at startup, and left it on until I shut down. I don't really want to do that, but since I use Firefox more than any other app, it's not the worst solution, I guess.
Sanglant, thanks for the suggestions, but I don't have any old PDAs. And I don't need the computer to start itself, I just need an application that will either start itself when a reminder is due, or will run very lightly in the background.
Bridgewalker, thanks, but I don't have Mac money.