Add a few drops of mild liquid soap into a small bucket of warm water. Dip a cotton cloth into the water and wring it out so that it's damp and not soaking. Scrub the chest thoroughly to remove any dirt and grime. Rinse the cloth often to unload the dirt you pick up and to re-wet it. Dry immediately with a clean cloth. Repeat this as needed until all the dirt is removed.
Pour a small amount of naphtha on to a clean cotton rag. Again, scrub the piece all over, pouring naphtha as needed. Naphtha is a solvent that removes buildup in the finish. If the dirt is fairly stubborn, you can switch to #0000 fine steel wool instead. Again, repeat this process until the dirt is removed. Wipe the piece down with a clean cotton cloth to dry it.
Cut a piece of cotton cloth into a 4-inch square. Wrap a small chuck of paste wax into the middle of the cloth and fold the corners in. Rub the cloth back and forth in the direction of the grain on the chest. This applies a nice even coat of paste wax all across the cedar chest. Having the wax in the cloth will prevent you from having to dip into the wax several times and distributes the wax evenly.
Wait about five to 10 minutes and then buff the wax off with a clean cloth. Keep buffing until there is no more resistance and the cloth glides across the surface.