Spread white toothpaste onto the permanent marker area. Do not use toothpaste with gel.
Rub in the toothpaste and allow it to sit for a few minutes. The products in the toothpaste used to remove tartar from your teeth go to work on the stain without removing the finish from the wood. However, if you forget about the toothpaste and allow it to sit for several hours, it can lighten the wood stain, so make sure to only leave the toothpaste on for a few minutes.
Take a damp cloth and rub some dish soap into it. Clean off the toothpaste from the cabinet going in the direction of the grain.
Thoroughly dry off the cabinet. The permanent marker stain should now be removed. Repeat the process if the mark is still visible.