Ok, the paper towel behind the thermostat solution only lasted a couple of days. I called service out again, they said they can't find anything wrong. They said they could replace the circuit board, but didn't see a need for that.
Mean while the furnace problem gets worse. It runs for 5 minutes, shuts off, then immediately turns back on, does this for over an hour. It's a Bryant 394HAD/396HAD Series A. It was installed in 1987, we bought the house 3 years ago with no furnace problems until this year.