David Marks uses 3/32" material that he resaws from heavier stock and then runs through a drum sander to smooth all the pieces and to get them to final thickness. He teached the double bevel method where the inset piece is placed behind the field piece, taped in place and then sawn with #3 blades (he uses Olsen 33 TPI) at a 6 degree angle. The field piece is any shape you want to insert into your finished piece (oval, circle, rectangle) which is template cut into the finish piece and then hogged out to just under 3/32" with a router. He recommends the Whiteside Template Routing kit (about $32) for the latter step. He also taught us the free-form method where his subject piece goes outside the template area and is embedded directly into the finish piece (vertical knife cuts around the inlay to define the imbed area and then hand routed with a 1/16" bit to depth).
Again, mucho kudos for your work -- I presume the whole finish piece (your Sapele backbround) is also veneer.