Basically, what was happening was that most of the experiments were sensitive only (or mostly) to one of the three types of neutrinos. It took until 2001 to collect enough heavy water to do the first observation that measured *all* types of neutrinos equally, at Sudbury Neutrino Observatory in Ontario.
But this post is not about that measurement. This post is about what you do if you have to find another way.
In June of 2001, just before SNO published its results in August, another one of these one-neutrino-only experiments (Super-Kamiokande, in Japan) released the results from its observations. While previous experiments had only confirmed the measurement of the neutrino deficit, Super-Kamiokande managed to provide evidence (albeit indirect) of the oscillations themselves by having the temerity to treat the entire Earth as nothing more than a part of its experimental apparatus (other experiments, like DAMA, do this now as part of the search for dark matter).
Why did this work? The reason electron, muon, and tau neutrinos oscillate between each other is that they are flavor eigenstates, not mass eigenstates. The flavor eigenstates are in fact built out of a superposition of mass eigenstates (the values of the matrix elements relating them is an active topic of research), and as they evolve in time each mass eigenstate picks up a time-dependent phase factor proportional to its energy. The upshot is that as the particle propagates along, the probability of it being in any particular flavor eigenstate changes. If you have a detector that can only see electron neutrinos, but two thirds of the electron neutrinos have evolved into the muon or tau neutrino state, you'll get only one-third of the neutrinos you expect!*
So what do you do if you can only measure electron neutrinos? Since the probability of a neutrino being in the electron neutrino state depends on its distance to the Sun, you just wait for the Earth to move closer or further away. You'd expect the flux of electron neutrinos to go up and down with the seasons.
And that's exactly what you get:
|arxiv link, PRL link|
There's something really special about this kind of magnificent simplicity.
*With some complications, but this is the simplified version.