To add to the excellent responses by WideRanger and Shinkansenfan, I'd also suggest the Hida Express from Osaka, Kyoto, or Nagoya to Takayama - a gorgeous ride through the Hida mountain range, close to the northern Alps. There are plenty of very pleasant places to stop off, including Gero (historic onsen town), Shirakawa-go (beautiful mountain village famed for it's thatched roof houses), Takayama (historic, mini-kyoto style city), and then heading up to the coast (albeit not on the Hida Express), Kanazawa which has some of Japan's best seafood, one of their most beautiful gardens, and is directly linked to Tokyo by the impressive Hokuriku Shinkansen.
The Toyama Chihou line from Toyama (where the Hida Express often terminates) to Tateyama in the northern alps is also really nice (mountains, rivers, and green fields). The cable car ride from Tateyama station to Bijodaira is cool too, good views from the top.