You have an old Honeywell with a variable resister meant to match the amp draw of an old fashioned gas valve. The new furnace may pull different amperages at different times of the cycle. A digital would be recommended as a sure cure for that portion of the problem. In the meantime you can open up the thermostat by pulling off the gold ring and there you will see what's called a heat anticipator that would normally be set to the old gas valves amp draw to keep it from short cycling. I think on a Honeywell you can move it to 1.2 on the scale at the most. That should be close enough if the thermostat isn't already out to make it stop the quick cycling. I'm assuming also that you put in a modern furnace with a glow coil or other ignition device but kept the old thermostat. Either way it sounds like your thermostat to me. Only a trained serviceman with a meter is qualified to work on gas appliances. This is just friendly advice.