I know the NDS has a timer (or timers) that can be set to automatically do something every so often. Does the Wii have anything like that? I want to be able to set a timer and then forget about it until I need to access the data that the callback function is updating (ie. I don't want to have to manually update the timer in every piece of code that may take longer to execute than the timer resolution, such as when you are waiting for input from the player).

I'm guessing I would either have to access the Startlet timer somehow (but isn't that an ARM processor? How would I write and compile separate code for that?) or do something with threading, which I have no experience in at all. :(

I just want something where I can say "every 15 milliseconds, run this function" (in code, of course).

Start a new thread, sleep 15 miliseconds, run your function.
If you need example code for new threads, just look at the sourcecode for the SDL Wii port.