As a grad student I took the train between Chicago and Mpls several times. That route ran through La Crosse and up the river.
One time I rode my bike back, favoring county roads. The flat roads along the river were too busy, especially on the MN side. So have I distinct memories of the hills on either side of the river. East of La Crosse I rode the Elroy-Sparta Trail, a relatively new creation at the time. Now there's 100 miles of designated bike trail in this area.
Elroy Sparta mapsMinnesota has designated a Mississippi River bike trail, with detailed maps. As long as you don't hit bike riders on the shoulder this might be a good resource for slow-and-local drives along the river.
Mississippi River Trail