Rail travel west coast of Japan

Can I travel by JR trains from Ueda to Hakodate via Niigata up the west coast of Japan? Is this a recommended route? I prefer rural and scenic rather that cities and lots of people so would this suit me more than travel by the usual route via Tokyo and the eastern side.

Can I travel by JR trains from Ueda to Hakodate via Niigata up the west coast of Japan?

Yes!

Is this a recommended route?

It is a very slow route. But it is scenic. So, if you do have the time, it could be a pleasant alternative to the endless tunnels of the Tohoku Shinkansen.

There are no major attractions along the way. But several minor ones, including Dewa Sanzan:
http://www.japan-guide.com/e/e7900.html

And Hirosaki:
http://www.japan-guide.com/e/e3701.html

The Gono line is highly regarded for its coastal scenery. They do run special tourist trains around there at certain times.The train is "Resort Shirakami", it runs from Akita to Aomori. Sorry but I am uncertain when. If you search the regular schedules on Hyperdia you will need to put in Higashinoshiro to Goshogawara via Fukaura to get Gono line trains.
This link may be of interest http://www.geocities.com/TheTropics/Cove/5750/scenic2.h...

Thanks for your help. How slow is slow? I have two days and one night to make the trip. I would love to stay in a small family run guest house in a rural area or small town. Any suggestions as to where to stop and stay? Travel time June 09. What will the weather be like? Rainy season? Rainier east or west?

This schedule is separate from what may be possible if Resort Shirakami is running.
Day 1
Depart Ueda 0636 on Shinano Railway to Nagano arrive 0715
The rest is JR. Nagano 0812-0941 Naoetsu 1010-1202 Niigata 1234-1615 Akita 1636-1739 Higashinoshiro at this point you get the choice of staying the night but I have no idea what is there or continue to Noshiro at 1800 and arrive there at 1805
Day 2
0755 from Higashinoshiro or same train from Noshiro
0802-1157 Kawabe 1400-1443 Aomori 1522-1733 Hakodate

You can check at http://www.hyperdia.com/cgi-english/hyperWeb.cgi

If Resort Shirakami was running you spend night 1 at Akita and take "RS" from there for day 2.

Check the schedules on the link I gave you. It is important sometimes to list passed points or the schedule shown will be the fastest and not the most scenic.