Disassemble the cabinets with your screwdriver, taking off all the hardware and laying out the parts on a work area.
Run a hand-sander over the each part of the surface that you want to cover with veneer, getting the surface flat and dull.
Measure the length and width of the first area you want to cover. Mark out those measurements on a piece of veneer with a carpenter's square and pencil, adding one inch all around.
Cut out the veneer piece with your utility knife along the marked lines.
Pull the wax paper off the back of the piece of veneer. Press the piece of veneer onto the surface with a one-inch overhand all around.
Cut off the excess veneer from the edges of the space, running your utility knife along the edges.
Wrap an old towel around a block of wood, so it doesn't scratch the veneer. Scrape it over the surface, pressing firmly, to get the veneer firmly set and free of air bubbles.
Repeat for each section of the cabinets. Put the cabinets back together.