Gas furnace motor cycling
After more than a month and service visits from multiple technicians from multiple companies, I finally got a technician who solved my problem. He listened to me describe what was happening and said "That's the thermostat. Let me see if I've got an extra one on the truck." 10 minutes later he drove off without charging me and the furnace has worked fine ever since. I feel like an idiot for not just installing a new thermostat in the first place, but at least I don't have to get up three times a night.