I can't remember who, and I'm at work right now so I can't look it up, but somebody in one of the TBB volumes suggested that plant fiber might actually be superior to sinew, because it does the same thing but is lighter in weight. He was thinking of flax and milkweed, if I remember right. I'm a little skeptical about that. If plant fibers were better than sinew, I think the natives would have been using it. Still, dogbane is strong stuff, so it should at least keep a bow from exploding, and maybe even increase performance. It's a lot easier to get than sinew, too--growing all over the mountains, free for the taking.
Sinew is more rubber band like: it can stretch easily 5% (wood at best 1%) but it's also more easily stretched than wood: its modulus of elasticity is 3 to 5 times lower than that of wood (it takes 3 to 5 times less force to stretch it a give percentage), while having a higher density. But the advantage of sinew is that you can stretch the hell out of it, and it shrinks as it dries, putting the belly under tension. It's advantageous in short bows where the back is strained far beyond 1%. If the back isn't strained >>1%, sinew is just dead weight making your bow sluggish.
Plant fibers on the other hand are much stiffer (modulus of elasticity of flax is 50 to 70 GP, compared to 2.7 GP for sinew and 12 for osage and 15 for hickory, just for reference), and although some plant fibers can stretch (allegedly) more than 3% (like sisal), they don't shrink like sinew as they dry.
It was Tim Baker who lauded flax as the ideal backing (for regular long bows), as it behaves like incredibly strong wood that you can mold as you like, and apply easily wherever you want.
But since flax is so strong in tension, it can easily overpower the belly of a bow.
So summing up, sinew and plant fibers can both shine or be awful as backing material, it all depends on the design of the bow.