Do they? Or do they often collide with atoms and experience the same kind of "conversion"? As far as I know, nobody has performed any experiments to find out. The very idea that they might change from one form to another is very recent.
On the contrary, we've been doing experiments about this non-stop for decades, and the answer is "no, neutrinos don't interact very much". While the interaction cross sections with things have kind of large error bars by particle physics standards, they're still known to ~20%, and are Really Tiny. A good perspective - the mean free path for your typical neutrino is something like a light year of lead before it interacts with matter at all, and when it does, it's not doing flavor changing. How do we know? Since we can't build a light year of lead sized experiment to catch half of the neutrinos we shoot, we build them as big as we can and shoot trillions of neutrinos per accelerator pulse, run the thing every couple seconds for years at a time, and observe those few neutrinos which are so incredibly unlucky as to smack something dead-on. How dead on? The weak force has a range of ~10^-18m. A proton is only 10^-15m in size. So a neutrino happily passes straight through a proton most of the time, to say nothing of all the empty space in an atom (which are 10^-10m in size).
On the other hand, the data fit the hypothesis of quantum mechanical flavor mixing quite well. That happens regardless of the presence of matter. However, if there is a lot of matter in the way (say, solar neutrinos exiting the core of the sun) it is a big effect - the "MSW Effect" explains how the presence of matter nearby changes neutrino oscillations.
In the case of the earth (which has a comparatively puny mount of matter) the effects are a lot more subtle, but neutrino beams going through a lot of the earth should be sensitive to this in the next decade or so. The Japanese beam is comparatively short so isn't ideal for such a measurement, the under-construction NOvA experiment in the US will do better, and people would really like to do a Fermilab->South Dakota beam to nail it down in spades.