I've seen listings for arrows made of anything you can get in your hands. There certainly nothing wrong with birch. I saw a maple arrow that was really nice and few oak. If you get a round dowel cutter, you can make shafts of any material you can put through it.
Alanesq, thanks for the spine tester link. It is great to have information on effective, simple, inexpensive archery tools.
Here is another one.
PA Spine Tester
For those like a digital touch, you can build this tester
Jim Hill's Spine Tester
Consider that spine testers, with some small tweeking, can be used to test for straightness. Quality multi-tasking is a sign of a valuable tool.