-- Last Updated: Jul-20-13 5:26 PM EST --
are probably the Current River and the Eleven Point. The Jack's Fork is probably doable from Alley Spring to Eminence at current water level. That would be an easy day trip. The Jack's will be too low above Alley Spring and I would not recommend going below Eminence due to water quality issues.
The Eleven Point will probably be too low above Greer Spring but you could put in at Greer and go down as far as you like (from Greer to Riverton is about 19 miles). At current levels there are a couple of rapids that might rate a Class 1+ or perhaps II- depending on your perspective.
On the Current you could run from Akers Ferry to Round Spring (again around 19 miles) or you could break it up into two days and camp at Pulltite campground, roughly half way between the two. There is nothing harder than Class I rapids on the Current and Jack's Fork, but one needs to be careful with rootwads which pose the greatest potential hazard to paddlers.
The advantage of the these rivers is that they are spring fed which generally keeps them at a doable level throughout the year. They are also very scenic. The disadvantage is that they can attract large numbers of livery customers on summer weekends, some of which can be noisy and sometimes unrefined. The Current and Jack's Fork tend to attract more of these folks than the Eleven Point. You chose to post this in the wilderness tripping subforum. Needless to say, you are not going to find any real wilderness river trips in the central Midwest but if you are looking for a river with a somewhat more secluded feel I would pick the Eleven Point over the Current or Jack's Fork.
There are plenty of gravel bars on the Current which make admirable campsites for overnight trips. The Eleven Point has fewer gravel bars but does have a number of paddle-in campsites. There are outfitters on the Current and in Eminence that will do a shuttle for you but they tend to be rather pricey. I would recommend running your own shuttle if possible.
They are really too low right now but for future reference some other regional favorites that are dependent on natural runoff are the North Fork River(actually North Fork of the White River) in southern MO and the Buffalo in Arkansas.
Overstock Outlet Foods
Canoe / Kayak Anchors