I haven't seen a pattern, but I'm wondering if you could get sheets of the plastic canvas grids that are used for yarn craft and just cut them to the dimensions you need for your boxes. They could easily be covered with fabric inside and out, and if you do the final seams with velcro along the top or bottom edge they could be removed for washing. Just be sure to pre-shrink the fabric before you begin. Cardboard would work, but the plastic would hold up better against rough use in a child's room.