It's not all Kobe beef, y'know...

I really like Hida beef cooked in miso on a layer of hoba (magnolia leaves) over a hibachi.... It's a delicacy in Gifu. There's a restaurant in the center of Shirakawa go that does this really well. It's not expensive... but who's complaining?

The butchers' shops in Takayama are also worth checking out. I recall one with one of the best displays of beef I've ever seen. They will happily arrange refridgerated shipments to any place in Japan. If you planned things well, you could do a tour of the country with choice cuts of Hida beef delivered to your lodgings every day just as you arrive...

Alternatively, you could tour the famous beef producing regions in Japan and eat local cow:
http://www.pulgogi.net/sanchi.html