Okay, using the +5v pin to trip the relays seems to be the problem. I used a 5v voltage regulator instead and it seems to have eliminated the reset problem. I guess I don't understand why I don't get reset issues when running off of USB power, because the board still has to supply the same 5v to the relays. Maybe with 12volts coming into the Arduino and 5v going out, its just too much power transfer for the board to take.
The real lesson to come away with is that the Arduino board is great at controlling things, but not so great at powering things.
Lefty