Here is an interesting survey that show the least cared about prefectures in Japan.  Here is the top five:

1. Shimane
2. Fukui
3. Saga
4. Tokushima
5. Tochigi

Check out Japan Probe for the complete list.

This link goes to the clickable link to each prefecture on the below map.  Shimane prefecture is number 34.

So what makes the fact that Shimane prefecture is number one on this list interesting you may ask?  Well thatis the prefecture that has been driving the whole Dokdo issue in Japan.