Most thermostats have a control to select either AUTO (fan runs only when heater/ac is on) or ON (fan runs continuously, regardless of whether or not the heater/ac is actually running. This switch is usually placed near the ON/OFF switch, if heater-only, or the HEAT/OFF/AC switch if multi-controlled.

Check to make sure that the switch is in AUTO mode. Again, this would mean, under normal operation, that the fan only runs when the heater is operating. If this switch is found to be in the ON position, that was your trouble. Switching to AUTO should solve your problem.

If, however, your switch was found correctly set (AUTO position), then your problem lies elsewhere. I would suggest replacing the thermostat with a new one, since they are fairly inexpensive. This would be a good time, also, to consider a programmable thermostat if your system will accept it. Make sure you buy the proper unit. There are models for Heat/AC and Heat only.

If, after properly installing the unit, your problem persists, it must lie elsewhere. You will probably need to have the actual furnace checked out by a professional.