Trace is best reliable bet.
You can ride the lake bottoms at Lake Sardis and Lake Enid when/if the water is low enough in the winter time. You can still run into "bottomless" creek crossings that the fat-tired 4 wheelers can cross that we can't.
Here is a link to some MX tracks: http://www.mxdownsouth.com/?page_id=9 Lone Lake MX used to be open outside Batesville, but I couldn't find their website anymore after a quick search.
This might give you some ideas for more dirt road oriented rides in Arkansas and on the edge of the Mississippi Delta: http://www.transamtrail.com/ Keep in mind that this website focuses on dirt roads NOT single track. A lot of the roads north of Sardis that are mapped are now paved.
There is some amount of dirt road (and a double track) riding in the Ozark forest north of Mountain View and west of Calico Rock.
A little farther east is Stony Lonesome Trails in Cullman AL. http://www.stonylonesomeohv.us/ Lots of fun and good for different skills levels. You could easily enjoy a weekend there. There are nice, if basic, cabins on the property.