Hwy 67 looks to run from St. Louis to Texarkana. I can vouch for the quality of this highway for cycling, at least from Texarkana to Little Rock. Adequate shoulder for most of the route and light traffic. Very nice. Use Google Maps, walking option to see it.
The problem is getting thru LR. Without going around the world, can't be avoided. According to Google, 67 is the only non interstate bridge over the river. There is an impressive bike bridge, but it is way on the west side of town and would require a significant detour down a picturesque bike trail that parallels the river. If you've got the time, worth a ride.
http://www.mapmyride.com/ride/united...rock/282355914
You could contact some
www.warmshowers host in the area to see if they could suggest a routing thru town.