I went last time to the traditional and old city Furukawa. The city is really Rural and we can feel the peace and Japanese old atmosphere. It's 20 min from Takayama by train. The train cost about 260 Yens. Contrary to Takayama, there are almost no foreign tourists in the city. Locals were really surprised to see us.
I rent a mountain bike with some friends. We had a tour in amazing beautiful route with a japanese guide who by the way speak english. The rental bike house is 7 min by walk from the train station.
I really enjoyed my time, and recommend to you to do so.