Hi,
All of your travels can easily be done on the bullet train lines - the Tokaido line between Tokyo and Osaka, and the Sanyo line between Osaka and Fukuoka. A 7 day rail pass covers all of it, and you can get some huge savings.
You don't give any info on your plans, but you are spending very little time in Tokyo and Osaka. Fukuoka has some very good places too, which you can see here, especially the Nanzoin Temple, Yusentei and Rakusuien Gardens, plus the newer Hakata City.
I would say if you can, to give more time to Tokyo & Osaka, and if you have never seen it, some time for Kyoto.
Have a great trip!