Set two 3/4-by-14-by-24-inch plywood boards next to each other on the working surface. Use medium-grit sandpaper to sand down the center of one of the boards, running lengthwise. Continue sanding until the middle of the board slopes gently downward, and will hold the plates in position. Consider this the bottom board of the plate rack.
Measure across top surface of the bottom board, making a mark every inch across the board, in a line two inches from what will be the front of the board. Drill a hole that is 1/4-inch deep and 1/4-inch in diameter using a 1/4-inch diameter drill bit and a drill.
Repeat the measuring and drilling on the second board so that it has the exact same hole spacing when the two boards are placed on top of each other.
Position two 3/4-by-14-by-14-inch plywood boards on their ends, parallel to each other about two feet apart. Have an assistant hold them in place. Set the two 3/4-by-14-by-24-inch plywood boards against their ends so that they form a 15 1/2-by-24-inch rectangular frame with their holes facing inward.
Drill three holes through each of the larger boards and into each of the ends of the smaller boards. Attach the boards with 2-inch wood screws.
Remove one of the long boards from the frame and position 14 1/2-inch dowels with 1/4-inch diameters within each hole. Set the long board back on top and push all the dowels into its holes as well to test the fit. Make any necessary adjustments to the depth of the holes so that the frame fits securely.
Remove the long board again and apply a bead of wood glue to each end of each dowel. Set them all in place once more and refasten the long board with its screws. Wipe off any excess wood glue.
Set a 3/4-by-15 1/2-by-24-inch plywood board onto the back of the frame. Drill holes around it and into the wood frame underneath, spacing the holes every six inches. Attach this board, which is the rear board, to the frame with 2-inch wood screws.