Choose wooden drawers with easy-to-remove faces, such as those held together with nails or screws.
Remove knobs or latches, and detach the face. If you'll be updating the hardware and will need holes in different places, fill old holes with putty, then sand smooth.
Cut a piece of fabric that is 2 inches longer on all sides than the dimensions of the face; snip the corners on a diagonal. Fold the cloth over the face, making neat corners and using a staple gun to secure as you go, stapling opposite sides first.
Apply matte acrylic medium (available at art-supply stores) to the fabric with a paintbrush -- this will soak through the cloth and adhere it to the face, and also add a protective, easy-to-clean coating. Reattach face.
To replace old hardware, cut a tiny X over each existing hole with a utility knife, and refasten. If you're adding new knobs, drill holes and attach where desired.