Reply by FellingStudio

My approach would be as follows (and, I think that I’m just paraphrasing what Shipwright said.)

1> Cut, dovetail, and fit the drawers. (Do all of the joinery.)
2> Lay out the drawer fronts as they will fit in the chest face down on your bench.
3> Using masking tape, tape the drawer fronts together. The aim here is to be able to maintain their positioning relative to each other throughout the remainder of the veneering process.
4> Flip the drawer fronts over so the surface that you will be attaching your veneer to is facing up on the bench.
5> At the moment, I’m a hammer veneering guy, so I would hammer my sheet of burl veneer on at this point.
6> Use a veneer saw to separate the veneer (and therefore the drawers.)
7> Trim excess veneer off of the edges of the drawer fronts.
8> Assemble the drawers.

The benefit to this approach is that you don’t have to worry about keeping the grain perfectly matched from drawer to drawer. It will come as a matter of course.

-- Jesse Felling - http://www.fellingstudio.com