In Robert Mussey's book "The masterworks of John and Thomas Seymour", Mussey indicates that white pine was the most prevalent choice for the ground of the veneer work. I was fortunate enough to meet with Mr. Mussey several weeks ago and have him authenticate a piece I purchased this spring at an antique shop in Florida. My piece is typical Seymour work and only the visable surface is veneered and the opposing surface is white pine stained to look like mahogany. That said, the pine is split and it shows through the face veneers.
If I were building a piece today, I'd probably want to stabilize the back or underside if I wasn't try to match an original. If you could access the book, it should provide you with the construction details you seek. Good luck! Ross