I haven't got any bandsaw, either. My first real homemade bow was a hickory copy of a Rapahannock Indian bow made entirely with chert scrapers. (Hey, if you're going traditional...) I'm working now on a red cedar selfbow for an enthusiastic four year old.
I recommend rough shaping a stave with a hatchet and rasp, then finishing by scraping with glass - it works just like stone, but is easier to get a regular edge on.
My question is, given that many of the old bows had nothing even remotely resembling a rest and many others did, how do you all decide which to make?