Wipe down the piece of furniture with a clean, lint-free cloth. Wipe gently to remove dust and other debris that could scratch the surface further.
Clean the wood with a commercial wood furniture cleaner. There are many spray-on cleaners available, although there are also some liquid products that must be mixed with water before you can use them. Read all instructions before using the cleaner on your furniture.
Pour a small amount of furniture polish that has either lemon oil or orange oil in it on the piece of furniture. As a homemade alternative, you can mix equal parts olive oil and lemon juice to make your own furniture polish.
Spread the polish over the furniture with a clean microfiber polishing cloth. Buff the polish into the wood using tight circular motions. Keep buffing the furniture until it shines but no longer feels greasy or wet. The finish will be dry and shiny.