Place a small dish of baking soda in each drawer. The baking soda is a natural odor absorbant that will also draw any moisture out of the drawers. Let the baking soda sit overnight.
Fill a spray bottle with white vinegar. The vinegar is a deodorizing agent and will also kill any mold or mildew growth that is causing the odor.
Spray the vinegar solution into all of the drawers.
Keep the drawers open, and allow the drawers to air out overnight.
Repeat if needed if all odor has not dissipated.