I wonder if the water traveling through a copper tube would setup some sort of measurable resonance that changes in frequency depending of flow and/or pressure...
Hmmmm...
Or maybe sound waves transmitted through moving water are displaced in the direction of the flow, or take longer to flow upstream with an increase in downstream flow. Sound also travels faster through cold water than through hot water, so there may be something to look at there as well.
Or how about magnetic field deviation based on flow...
None of these methods would inhibit flow or pressure, and the hardware could be mounted on the outside of a clear vinyl tube. I'm not an engineer by any stretch of the imagination (but I did stay at a Holiday Inn Express last night), but I'm happy to throw a grenade like this in the room and then walk out.
P.S. If anyone does come up with something based on one of these ideas, I have to insist that you call it the "TurkeyMonkey Flow Meter", in honor of my avatar.