Jeff, when you turn on the thermostat, that turns on the fan. If the fan air flow is sufficient to trip the sail switch, it is supposed to turn on the gas valve and the ignitor. When this happens, you should be able to hear the clank of the gas valve turning on, and depending on conditions, you might be able to hear the snapping sound of the ignitor.
If the thermocouple in the burner does not sense the right amount of heat, the board will time out and shut off the ignitor and the gas valve. It will NOT shut off the fan, so the furnace will just blow cold air.
We have seen the symptom that you describe. In our case, the furnace would always light again after the fan stopped, but not before.
Low voltage at the furnace will cause boards to malfunction and to fail. Low voltage does not prevent the fan from running. Measure with a DVM. Voltage should be 12 volts or very nearly that for normal ignition.
If you decide that the furnace needs a new board, I recommend that you do not use the manufaturer's replacement. Dinosaur boards are much more reliable and cost no more than OEM. Current models include a fan shutdown feature for a little more money.
Try this link: www.dinosaurelectronics.com/
Good luck with your furnace.