TO THOSE WHO ARE GOOGLING IN YEARS TO COME: Here's what we've learned so far: Something in the furnace went bad. The 12v power source for the thermostat goes through the circuit board of the furnace. If we jump across from the main 12v fuse box to the power supply on the thermostat, the AC kicks on normally. The circuit board on the furnace should have a flashing indicator light, but it isn't working, so probably we're looking at swapping out just that circuit board, or (more likely) putting in a permanent bypass from a fuse in the box to the thermostat. I'll update if we come up with a different diagnosis, but it's looking like that's it.