Unplug and remove the fountain pump.
Rinse the pump with white vinegar, and scrub it thoroughly with an old toothbrush.
Rinse the pump with a garden hose.
Drain the fountain according to the manufacturer's directions.
Pour 1/4 cup liquid bleach into the fountain. Scrub the inside of the fountain thoroughly with a scrub brush, adding more bleach if necessary.
Rinse the inside of the concrete fountain thoroughly with a garden hose. Any bleach left behind can hurt birds or animals that stop by for a drink.
Replace the pump, and refill the fountain with clean water.