Whether unwanted artistic expression or accidental marking, marker on wood surfaces can present a removal challenge. Unfortunately, marker that's allowed to remain can soak right through the layers of finish and through to the wood. Try to address the marks promptly with these simple tools. Keep in mind that wood types and surface finishes can vary widely so even the most basic methods should be tested prior to use on a larger area.
Take a clean soft rag and dampen a corner with naptha. Rub the naptha-dampened cloth directly on the stain, along the direction of the grain of the wood. Naptha is safe to use on a range of finishes, but you should spot test to be sure. If application of naptha causes clouding or softening of the finish, try an alternative solvent such as alcohol or otherwise according to your best guess as to the finish type. After using naptha to remove the marker stain, follow with a light conditioning using wax, oil, or the appropriate polishing compound.
Caution: Never mix cleaning agents or chemicals, the result can be dangerous or deadly. Before cleaning, always test the agent on an inconspicuous location to determine its suitability and to make certain it does not damage the material. Wear appropriate clothing such as gloves and protective eyewear, and work in a well-ventilated area. Accidental inhalation or ingestion of cleaning agents can be hazardous and even fatal, particularly to pets and children.