Cut the plywood board into two equal-sized squares. This will determine the height and width of the rack and can be made to any desired size.
Draw an X of equal size on both boards with a pencil and ruler, from corner to corner. The the width of the X should be a few inches thick, but again, the design can be made to any size.
Cut the X's out of the plywood with a saw. Since the X will be untouched, it is really the extra wooden triangle shapes that are being removed. Drill a hole at each peak of the triangle to make sawing the angle easier.
Sand the pieces so that all of the edges are smooth.
Clamp both of the X's together securely with the edges lined up so that neither will be able to move. Insert a drill bit into a hand drill to make holes big enough to fit the dowels; although the recommended dowel size is 3/8 inch, any size dowel can be used.
Drill one hole in the center of the X's and two more 1 to 2 inches from the bottom of the X's. Drill a hole 1 to 2 inches from the top and continue to drill holes 1 inch apart until reaching the center of the X's.
Unclamp the X's and stain, finish, paint or spray paint both to any color. Make sure to prime the wood before painting, and coat the surface evenly to prevent warping. Allow the wood to dry completely.
Determine the length desired for the rack, though 13 1/2 inches should be a perfect size to hold newspapers. Count the number of holes in the X's and cut that number of dowels to size.
Glue the ends of the dowels into the X's and allow it to dry.