Furnace won't cycle

[deck52]

I'll have to describe a switch (because I don't know what this switch is called) before I describe the problem. There is a conduit box on the wall next to the furnace. There is flexible conduit going from this box to the furnace. On the conduit box cover is a switch, and on top of the conduit box is what looks like a transformer. Attached to the transformer are the wires that go to the thermostat upstairs.
Here's the problem: sometimes the furnace won't come on even though the room temperature is well below the thermostat setting. If I turn the switch described above off then back on again, the furnace kicks in immediately. Sometimes I can go for weeks without the furnace doing this, and sometimes I have to turn the switch off and on several times a day. I have replaced the thermostat more than once, and replaced this switch once too. The problem persists. Help!

[HayZee518]

turning the power of then on is resetting the internal computer. trouble shooting it to find out why its cutting out is another story.