Unscrew the hinges that hold your cabinet doors on their frames. Remove the screws with a screwdriver so the doors come free, leaving the hinges still attached to the cabinet frames.
Cut out the elements for your veneer inlay using a craft knife. Place each piece of veneer over a piece of scrap lumber to protect the surface you're working on and give yourself a soft material to cut into. Cut the veneer freehand or make straight lines using a craft knife with a metal ruler as a guide.
Lay pieces of veneer that are already cut over new pieces you're cutting to use the previously-cut pieces as stencils while you work. Make pencil marks to guide you while cutting. This will help ensure that each piece fits perfectly with the pieces next to it.
Assemble your final inlay on the cabinet door. Make sure there are no gaps or overlaps between pieces of veneer. Trace the edge of the entire design carefully with a pencil.
Remove your inlay one piece at a time, assembling it on a nearby tabletop or other surface.
Cut away the cabinet door's veneer surface with a craft knife using the pencil outline as a guide. Peel off the veneer once the entire outline has been cut, using the craft knife or by gripping loose edges of the veneer between your fingers and peeling it away.
Apply a thin layer of veneer adhesive to the exposed wood below the cabinet veneer. Press your inlay veneer pieces into their positions on top of the adhesive. Allow any adhesive that rises between pieces of the inlay to remain.
Place a piece of wax paper on top of your cabinet door, covering the entire inlay design. Place a sheet of felt over the wax paper. Lay a piece of scrap lumber with a smooth surface that's a similar width to the cabinet door on top of the felt.
Place another piece of scrap lumber against the back of the cabinet. Clamp the entire package together with metal clamps spaced several inches apart. Allow the adhesive to dry overnight.
Remove the clamps, lumber, felt and wax paper to reveal your inlaid cabinet door.
Sand down the inlay with 120-grit sandpaper until all of the veneer surfaces are the same thickness and form a smooth surface that's flush with the old door veneer.