Place two 3/4-by-24-by-36-inch plywood boards on the working surface and have an assistant hold them parallel to each other. Arrange them on their long, thin sides about 2 feet apart. Think of these boards as the side boards of the dishwasher cabinet. Position a 3/4-by-24-by-24-inch perpendicular to the side boards and pressed up against their ends.
Drill holes through this board, which is the top board of the cabinet, and into each of the side boards. Space the holes every 12 inches. Use 2-inch wood screws to attach the boards together tightly. Repeat on the other end of the side boards to complete the attachment of an identical bottom board.
Lay a 3/4-by-24-by-37 1/2-inch plywood board on the frame evenly. Drill holes every 12 inches around the board and into the top, bottom and side boards. Firmly screw down this board, which is the rear board, using 2-inch wood screws placed in each of the holes.
Paint or stain the cabinetry, if desired, by sanding all visible surfaces. Add a coat of paint or stain with smooth, even strokes in the direction of the grain of the wood. Allow two hours for the first coat to dry, then apply additional coats until desired coloring is reached. Repeat for three coats of sealant with the same procedure. Allow cabinet to dry overnight before installing.
Position the cabinet next to the dishwasher and mark the location of each tube or cord that must exit the rear of the dishwasher through the rear board. Drill a hole for each exiting piece, cutting out larger pieces with a saw if necessary.
Set the cabinet with the other kitchen cabinets a few inches from its final position. Connect the tubes and cords of the dishwasher through the holes in the rear cabinet, if possible. Slide the cabinet all the way back to its final position and set the dishwasher within it. Adjust the cabinet further to achieve the desired position.
Drill holes through the insides of the adjoining cabinets and into the boards of the dishwasher cabinet. Attach the cabinets together with 1-inch wood screws, repeating as needed for additional stability.