There are a number of rivers in Wisconsin that might fit the bill. I’ll mention just two right now.
The lower Wisconsin River is flatwater, but with a steady, sometimes quite brisk current, innumerable islands and expansive sandbars if the water isn’t unusually high (if the water is high, camping opportunities can be few and far between, but this can be monitored via the USGS gauge at Muscoda). This river can be sort of crowded on summer weekends, but gets more and more empty as you go downstream of Spring Green, where seeing just one or two small paddling groups over the course of a couple days is pretty typical. Most parts of the river show no sign of man-made structures.
If you prefer some whitewater, one popular option is the North Fork of the Flambeau River. It’s a good river for that north-woods feel. A good chunk of the North Fork is within a State Forest that has several small campsites (first come, first served). The rapids vary from easy riffles to slightly challenging when the water is low, but when the water is high, some rapids are likely to be a bit too much for average paddlers, especially when carrying a load of camping gear. There’s a USGS gauge but it’s below the last dam on the North Fork and not very useful for determining river level on the more popular and scenic sections, but a call to any of the outfitters should give you a good idea what kind of water conditions to expect. Check out http://www.paddleflambeau.com/