Names of the rural city areas?

I would like to go see some of the country side in japan.

There are thousands of "rural cities" in Japan- you will need to be a bit more specific! What kind of things would you like to see?

farms, old small school houses, large open areas, lage old japan farms

There are plenty of farms and rural areas all over Japan, even here and there around the Kanto, but the highest percentages are in northern Honshu and Hokkaido. Almost anywhere you go there are lots of fields. The largest farm in Japan is the Koiwai Farm in Morioka.

As an option, Kochi Prefecture on Shikoku is also totally rural. Agriculture makes up about 30% of the prefecture's economy, the highest in Japan outside Hokkaido.

You might want to check the Hida area in Gifu (Takayama, Ogimachi, Ainokura), where there are many old thatched buildings preserved. The train ride to Takayama from Nagoya is also scenic following a river gorge.

As far as very large open areas and large farms are concerned, Hokkaido is the best. If you particularly want to see dairy farms, then go to Obihiro or Tokachi while Biei and Furano are nice and popular areas for large flower farms though Biei I think will be better than Furano that is kind too touristy. I don't recommend the Koiwai Farm in Iwate as the dairy farm is also too well-known sight-seeing spot.
For reference: http://www.excite-webtl.jp/world/english/web/?wb_url=ht...

But when it comes to large old Japanese farms, I think they must be those with paddy fields, well seen like in Ohsaki in Miyagi or in Niigata.