I use Black Linen for a backing on the first bow I built 3 years ago, This bow is a red oak board bow. The bow had a lot of run off in it.
Still shooting it every weekend, pulls 55 @ 28. 68'' ntn , But like TommanyKnots said,,, make sure you size the back of the bow.
Also I came back over the top of the Linen with a thin layer of glue. Just make sure you give the glue plenty of time to cure.
Linen and TB III in my opinion makes a very durable backing. Good luck and put up some pics when you are done.