I can imagine the noise is larger than the signal for small drops. If you're intent on using piezos,
and you've already tried a lot of different amplifiers, well there are many different piezos
on the market, so you might just buy one or two of every one you can find and test it.
I imagine piezos pick up background sounds and vibrations, so possibly placing the pickups in
some kind of sound-deadoning box might help.
OTOH, if the noise problem is just be too great, well raindrops are water and should have some
conductivity, so you might try going to some kind of conductivity-sensing device.
A couple of other ideas also come to mind. One, is to build a little reservoir so the raindrops
will "integrate", so one large drop will flow out the outlet when enough small drops come in.
A simple funnel might do, as the drops will tend to congregate due to surface tension and a
larger drop will drop off the bottom of the funnel. Plop!
The other idea is some kind of raindrop light-beam interrupt pulse-counter.