Very catchy thread title!
Japanese can't understand each other?
No, in the sense that there are many dialects in different regions of Japan.
But also, yes, they can, in the sense that many dialects are now very much influenced by the so-called "hyojungo" that is used on TV and other national media and are rapidly losing their distinctive and colorful flair.
The influence of Kansai dialect(s) is also strong, causing inflection and accent changes in what is supposed to be hyojun-go that young people speak.
Today, strong dialect differences can be heard only among the older generations of Japanese.
Regarding Tohoku-ben (of which there are many variations), I'll write only what I know, since I'm not familiar with the dialect(s). AFAIK, Tohoku-ben retains the color and vocabulary of old Japanese and is very distinctive when spoken. I think the dialects of the Tohoku region are a interesting subject to study if you're interested in Japanese linguistics.
Coming to where you are going to live in Japan, I think you won't have any problem living in Sendai and going to Tohoku University. The city itself must have its own dialect, but it is a very large city where hyojungo (or its local variation) should be understood by (nearly) all. Plus, Tohoku University is a national university and a top-ranking one in the field of physical sciences, drawing students from all over Japan. The "common" language should be hyojungo.
The only dialect that is extremely difficult to understand for the non-native may be Okinawan, because of its distinctive culture and separation from the rest of Japan both in history (occupation by the US until the 1970s) and geography (ocean). However, many locals understand hyojungo, so language isn't a big deal there, either.