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.