An anonymous poster criticized my use of LM317 chips as current regulators for high-power LEDs. While I think they are great for 350 mA single LEDs, they are clearly inappropriate for the 12 watt LED that I was playing with.
My LED was taking around 800 mA @ 12 volts. A little searching around reveals that a good alternative is the AP8803 which is a LED buck driver which can handle 1 amps @ 8-30 volts. The nice thing (aside from its 92% efficiency) is that you can set up a dimmer circuit on it using minimal parts -- or PWM control from a microcontroller.