Found a few links.
I don't know where you're located, so these may be way out of the way.

Some look more like the small cabin trips mentioned above, but now I'm fully locked on the idea of staying in a yurt or something similar. <img src="/images/graemlins/smile.gif" alt="" />

Aspen Expeditions

Colorado Hut to Hut

Huts.org

Appalachian Mountain Club

Alpine Club of Canada

Sawtooth Guides (Idaho Yurts)

White Mountains, NH

The Colorado Rockies seem to be where the most hut to hut opportunities are in the US, but there are some on the Appalachian Trail in the Eastern US, some in Alaska, and others spread out across the country depending on terrain and season.