The decay chains of U-238 and Th-232 produce electron anti-neutrinos. These are the geo-neutrinos so-hotly disputed in this thread.
But the background level of anti-neutrinos is mainly due to beta decays in nuclear reactors. These may be detected via the inverse beta decay reaction, at a threshold of 1.8 MeV.
In fact i think a directional detector for electron anti-neutrinos is possible, because the “delayed coincidence signature” should indicate momentum, but i’m not sure whether this is resolved in current detectors.
— http://en.wikipedia.org/wiki/KamLAND
Yes, you’ll be detecting a very small number of neutrinos. Most neutrinos passing through the Earth emanate from the Sun, (and trillions pass through the human body each second). and there are also cosmic sources.