When I was a kid I used a brown paper, two-fletch system on all my arrows. I cut up paper bags from the grocery store. They worked OK as long as I crinkled the paper really well before attaching to the arrow. A three fletch quickly become a two fletch anyway... so I stopped mounting three.
Birch bark should work fine for a few shots as long as the fletching is damp or wet. Cut some "vanes" into the front part of the fletching so you only loose a little bit of the fletch each time you shoot... or the entire fletch will tear off at one time.
The best non-feather material I've used for fletching is the top of ziplock bags. The "zipper" part becomes the quill. The best natural non-feather material is fur.