There are '00s of places that fit that description between Tokyo and Hiroshima, provided you're willing to travel a bit away from the main line. Here's one example - pretty random, just searching for places that aren't huge concrete mega-resorts, that offer a private bath, and that are located within "reasonable" striking distance of the Sanyo Shinkansen. (What's "reasonable" to me might not seem reasonable to you. YMMV as we say).

Migawatei Ochikochi　(Shore-side inn - Ochikochi)
http://www.ochikochi.co.jp/en/

With the JR pass, you could take a Hikari train from Tokyo to Osaka / Kobe / Okayama and change to a Sakura train to Fukuyama. If you arrange in advance, the hotel shuttle will pick you up there.

The neighborhood around the inn has a few old buildings and cobbled streets. The location looks like a good stopping point if you're planning to visit Hiroshima, Kurashiki or other sites in the region:
http://www.japan-guide.com/e/e3432.html