I just want suggestions on places to live with a low cost of living.Its good to do this kind of research before applying. I'd recommend applying for prefectures that are just adjacent to major areas. For example: Chiba, Ibaraki, and Saitama put you within close range of Tokyo without the expense of living in the city. Also, if you have a good reason for choosing a prefecture you will have a much better chance of actually getting placed there.
But be aware that just because a place is not a big city doesn't mean that it isn't an expensive place to live. It really depends a lot on the particular location, for example in my prefecture it may actually be cheaper to live in the largest city once you factor in transportation costs, entertainment, etc. The people who end up spending the most seem to be those that are farthest from the big cities. Their rent may be low, but everything else is the same and their transportation costs can be much higher.
Also, I think you are mixing up modern with urban. Everywhere in Japan is modern. There's even toilets on the top of Mount Fuji. If you are looking to have the lowest living costs I would look at places that are outside major city centers but still connected by a good train line. Rural gets expensive due to poor public transportation, and a car is going to wreak havoc with your budget.
I am a person who likes to eat good food and go to entertainment places for cheap, but I think that will be difficult in big cities.
Not necessarily so. Generally speaking you can find good food and cheap entertainment anywhere in Japan, but big cities have more variety which can save you money.
Ya they extended it to 5 years now, used to be 3 though.
This is not entirely correct. The maximum extension is up to 5 years, but your contract isn't automatically renewed up to that point. Upon acceptance to JET you can get a 3 year visa and your contract will be extended up to 3 years assuming you don't do something illegal. The last two years, however, are conditional and you can be turned down for an extension. An actual requirement is that you have made an effort to learn some Japanese.
Basically, don't count on 5 years. Just be aware that although its rare that your extension is turned down, it does happen.
Anyway, Sira's advice is right on the mark:
In any case, as they say in JET, ESID- every situation is different- you might end up in a place with a low cost of living but where you have to pay all or most of the rent yourself, or you might end up in a more expensive area where your rent is free, meaning you can save more money.What matters more in my opinion is your lifestyle- you can save anywhere you live if you make the effort to.
And let me add that you should choose places to live because they are near places that are interesting and that you want to visit. That will save you money.
Other than that, don't worry about your placement too much. Learn how to live a frugal lifestyle in Japan and you can save money anywhere you're placed. The important thing is to enjoy your time in Japan wherever you are.
Good luck with your application!