Japan's "three most beautiful gardens" are
Kenroku en in Kanazawa (quite a shlep from your route)
Koraku en in Okayama (ditto)
Kairaku en in Mito (do-able but nobody ever talks about this one... wonder why)

I've seen Kenroku en and Koraku en. Both beautiful but too full of tourists to be able to relax. Actually I found a photo of Koraku en on Wikipedia which was taken by jpatokal. I wonder if that's the same jpatokal...
Nah - probably someone else with the same name:
http://wikitravel.org/en/Image:Korakuen_Overview.JPG


Tokyo

My favourite formal gardens:
Shinjuku Gyoen
Koishikawa Korakuen (Near Tokyo dome)

Near Tokyo - Kamakura
Very easy day trip from Tokyo. Get off the train at Kita Kamakura and walk into any of the temples around there, then wander down towards Kamakura. Just superb.

Near Tokyo - Yokohama
Sankei en
(Could be combined with the Kamakura trip)

Kyoto

Katsura villa

Lesser known gardens:
http://www.kyotoguide.com/e-past-issue/10ts.html#garden

Near Kyoto - Hikone
A very half-day trip from Kyoto.
Castle town with an original castle (one of Japan's best three castles!) and beautiful garden nearby - I forget the name of the garden.

Near Kyoto - Nara
Nara is a really green city with huge park area and some temple gardens. Really excellent day trip from Kyoto for garden enthusiasts IMHO.

Japan
Best gardens in Japan:
http://www.japan-guide.com/e/e2099.html