I was able to find large plastic moving boxes -- 20 inches by 24 inches by 12 inches -- from U-Haul for $1.50 a piece. U-Haul doesn't throw in the dolly, but that's only $7 to $12 more to rent.

Large Plastic Boxes. Sterilite 64 Qt Latching Storage Box, Stackable Bin with Latch Lid, Plastic Container to Organize Clothes in Closet, Clear with White Lid, 12-Pack ...

Start with a 66-quart bin that's 23.5 inches long by 16.5 inches wide; the length of the bin is the height of your drawer set. Gather up a collection of upcycled cardboard boxes or stack of foam ...