Pete, this might surprise you, but there is an even more effective way to use that 2x4 and that strap. (Check YouTube for videos, this is somewhat common in Russia for tractors.)
Mount the 2x4 along the face of the wheel, with about 2" or 3" of the board protruding on both sides of the tire. Too long and the board would hit the wheel well, of course. Strap the board tightly to the wheel.
This works with loose gravel and mud and snow, but not with hardpack. Do it on both sides of the truck.