Another easy route is Rowe Mesa which runs parallel to Interstate 25. There are miles and miles of dirt roads to explore on the Mesa, including several look out areas with incredible views to the North.
On the map below, this is the way to the top of the Mesa. From here (red marker), you can drive any direction you want. There are several gates, which you can open. If you do run into any private property, the gate will be lock and/or sign posted.
The above photo is farther west of Rowe, NM.