The Eastern Circuit is composed of the following four subprefectures:
The most sparsely populated region in all Japan, this vast area is nearly 40,000 km² in size. While the easternmost part is largely flat, rural and filled with endless cow pastures, much of the circuit is ringed with mountains.
Here, if anywhere in Japan, a rental car will come in handy. Train coverage is severely lacking (many lines have been terminated due to a lack of customers) and buses are infrequent, inconvenient and expensive.
Eastern Hokkaido has beautiful coastline, wonderful outdoor hot springs, and remote hiking. Some combination of these should make for a satisfying visit.
Hokkaido on the whole is good for seafood — if you're on the coast — and dairy, and the east is no exception. In central Hokkaido, near Asahikawa and Furano, there are a plethora of melon stands, though by foreign standards, Japanese melons are exceedingly expensive.