You will need 1x3 boards, dowel rods, and a few tools to make your own.
All the dowels need to be cut perfectly equally. The best way to do this is to set up a stop block on your miter saw and make all the cuts.
A Forstner bit is the best way to make holes in your 1x3s for the dowels.
Add wood glue to the dowel holes and add the dowels.
Clamp together to ensure it is square and allow the glue to dry.
Install drawer slides to your cabinet and and install the drying rack drawer.
Paint your drawer front to match your cabinets and attach it to the drawer.
That's it! Your drying rack drawer is ready to use!