How can I diagnose a superdrive that won't read disks?

My MacBook (Snow Leopard) seems to be unable to read DVDs and CDs. When I insert a disk (DVD or CD, I've tried a couple of each), the drive spins up, and (it sounds like) moves the laser back and forth a few times, then ejects the disk. I haven't tried blank media.

I've searched around the nets, and it looks like I may be experiencing a relatively common problem, but I don't want to jump to conclusions. Unfortunately, all the threads I've found are filled with "me too, me too", and not a lot in terms of actual help.

What tools can I use, or tests can I run, to try to diagnose (and perhaps even solve) the problem myself before I give up and replace the drive?

Try running the Apple Hardware Test. If I remember right, reboot holding D. If that doesn't work, reboot while holding D, with an OS X disk in. Of course you'll need to have an external DVD drive for that to work.

It would depend first on what it means to not read the discs. Does the disc insert and then eject, or simply not show up in the Finder? Does it mount and not burn properly? etc.

You can check for the optical drive in your System Profiler to see if the hardware is present (see: Disc Burning).