I am assuming you have a DuoTherm thermostat and air conditioner with the analog thermostat. The click you hear is the relay in the ceiling unit activating- this is good.
There should be 2 blue with white stripe wires coming out of the ceiling box- either both will be hooked to wires to the furnace, or one will be hooked to the furnace, the other to 12 volt power (some use one wire from the furnace to power the board as well).
Basically, the point is when the thermostat calls for heat, the board feeds 12 volts to the thermostat lead at the furnace- you need to find where the power is stopping. Could be a fuse on the board, could be a bad connection.
-- Chris Bryant