I have not done alot of hiking in Japan, but I know a few people who have done some, and I've travelled over 70% of the country, and have noted where they go. If you get a shinkanzen or another express train on the weekends, you will often see hikers going off into the mountains to hike. You see it particularly around Tokyo. I guess the best places to hike are considered:
1. Hokkaido - Actually this is one of the two places of significance I have not been in Japan. But I will get there. This area is highly regarded for its mountaineering during the summer.
2. Matsumoto - This alpine area is near Nagano, central Japan. Its easily accessed from Tokyo, Osaka or Kyoto by express trains on the Chuo Line.
3. Minakami - This town is on the Tokyo-side of the ranges. This is a great area for hiking, canoeing and canyoning, and there are several commercial operators in the area. A few stations after Minakami on a local train, there is a station that hikers alight at, which is actuall underground. Not a normal phenomena in rural areas, but 'very Japanese'. Basically when you climb out of the station exit, you will have already started your climb to the summit.
4. Mt Fuji - Alot of westerners and Japanese seem to like the idea of climbing Mt Fuji. Not my style. They climb up at night so they can be there for the 'rising sun'. I frankly dont like the idea of being in a cattle herd, even if in the day time. You would probably want to get a bus there, as I believe you want to get 1/3 of the way up by road.
5. Okutama - This is a mountainous area on the westen fringe of the Kanto plains (Tokyo). You can get a train to Hachioji or Ome, and you get the Ome Line to Okutama Station (the end of the line). From there, you can get a bus to Okutama Reservoir. We walked back along the river on a very nice trail. This walk is particularly attractive during the Cherry Blossom season and maybe even nicer during the Autumn fall.
6. Chichibu - Not sure where people walk in Chichibu, but there is an express train there from Tokyo.
Its fair to say that these areas are popular because they are readily accessed from Tokyo, but having travelled around Japan alot by train and glanced over many online, free, downloadable 1:250,000 maps of Japan, it strikes me that there are many other nice areas for walking. Hopefully in time I will get those notes and photos on the internet.