I spent about a month in Shimane staying with a Buddhist priest and his family to learn more about Jodo Buddhism. There's not too much in Shimane (always competing with Tottori for the least populated prefecture), but I absolutely love it there. You can always go to Matsue for the castle there, and then go to Izumo for Izumo Taisha (Izumo Grand Shrine), which is quite beautiful. Quite near to where I was staying is Iwami Ginzan, the silver mines that produced much Dutch silver during the Tokugawa period. As it were, the priest's main temple Shogen-ji is also located in Iwami Ginzan. IG is also home to lots of old shrines and some pre-modern ruins. In Nima (in-between IG and where I stayed on the coast) is the Sand Castle Museum, which I have yet to go to actually. You can usually take an overnight bus from/to Izumo to/from either Tokyo or Osaka.