What part of Michigan?  The UP has not had alot of rain as of late and rivers west of Delta/Marquette counties are fed mostly by run off.  The Menominee holds her own through the year, has enough length for a 2 day trip and has several (unpublished) campgrounds scattered about.  I would not drink the water due to industrial discharge.  The Fox/Manistique system and the Tahquamenon in the eastern UP have ample flow through the summer because of ground water and spring contribution.  Much of the watersheds to these rivers are public or commercial forestland and state forest campgrounds abound.  Water is drinkable if you filter.  The Escanaba system is very nice, but in my opinion should be run in May/June.  Water levels drop very quickly after the spring run off and the main river widens out over shallow riffles too numerous to count.

Don't rule out day trips on smaller river systems while car camping between runs, i.e. Autrain, Indian, or Two Hearted Rivers.  These can be free from housing development and are more enjoyable. The UP offers much solitude when you get off the beaten path.

Only a sampling here, I'd be happy to share more detail if you need specifics.  

-YB