Wipe fresh dog drool off the wood using a damp microfiber cloth. The cloth picks up all the drool and wipes it away. Regularly dust the wood furniture using a damp microfiber cloth to remove any drool or other allergens you can't see.
Spray an enzyme-based cleaner on the dog saliva, and wipe it up with a soft cloth. The enzymes break down the proteins in the saliva and successfully gets rid of the visible drool, as well as the allergens in the protein.
Vacuum the drool with a vacuum that has a HEPA filter. This collects the allergens from the drool and disposes of them without recontaminating the air. Vacuuming works best on old, dried drool.