I don't remember how it came up tonight, but somehow I ended up looking at the pendant Drew built back in October out of UV LEDs and a uranium glass marble. It's a very cool project, and I urge you to go check out his photos and video of it in action (and schematic!).
The electronics in it form what is called a ring oscillator. A ring oscillator is an odd number of inverters connected in a circle. Each inverter outputs the opposite of its input, but since there is an odd number, it's never stable, and will oscillate. Its frequency depends on how much you slow down the signal by using resistors and capacitors. If there was an even number of inverters, note that it would be stable, which isn't useful for an oscillator, but can be used to store information, and is known as SRAM.
Looking at Drew's schematic, I realized that with a little digging, I probably had all the parts to build the same oscillator, though with only three stages instead of seven. I also don't have any uranium marbles or UV LEDs sitting around, so we'll have to settle with the significantly less awesome bright red. I've tried to build this circuit before using a 7404 inverter instead of discrete components, but didn't quite grasp the concept at the time enough to get it to work.