Fill a bucket with 1 gallon of warm water, and mix in 1/2 cup of trisodium phosphate. Put on a pair of rubber gloves to protect your hands. Agitate the water to break up any clumps of the powder. Stir the water until the trisodium phosphate is fully incorporated into the water.
Dip a cloth into the solution, and wring it out so that it is just damp. Wipe up some of the mildew and rinse the cloth off in the solution. Repeat the process until most of the mildew is gone. If you have difficulty removing the mildew with just the cloth, use a soft-bristled scrub brush to lightly scrub the wood.
Fill a spray bottle with white distilled vinegar. Set the bottle to a fine mist setting. Spritz the finished wood lightly with the vinegar, and let it sit for about 5 minutes.
Clean out the bucket, and fill it with clean, warm water. Dip a clean cloth into the water, and wipe up the vinegar. Open any windows in the room. Place a fan near the wood to help it dry. If the finished wood is part of a piece of furniture, set it outside to dry in the sun.