KOA campgrounds quite often have the small cabin things, as well as many former KOA campgrounds. Unfortunately most of them are usually near interstates and not necessarily the back roads that most touring cyclists prefer. In the Midwest in particular many small towns will allow camping in a city park if you ask first, town halls are a good place to ask. Here in NC small country churches, and VFD (volunteer fire departments) are good bets, again... ask first. On many occasions I have asked at a local country corner store and have been directed to someone that knows a place to camp and in many cases has offered me a bedroom at their house or a camping spot in their side yard.

Aaron