One of the best robots I got to help design was made of plywood. Our chassis design used various thicknesses of baltic birch plywood (1/2, 3/4) depending on the load seen by each frame member. Some lower quality 1/4" plywood was used to brace the frame members. All were held together with good wood glue and long staples. We were fortunate enough to have a shop with good saws to get all the pieces cut precisely, and a pneumatic staple gun. 2009 was a year with low loads on the drivetrain, so this was a great material.
http://www.chiefdelphi.com/media/photos/32671
To my knowledge this is the only robot built with 2x4s and plumbing supplies to win a quality award. Krylon paint works wonders!