If you have a bit of time, there are zillions of dirt roads and Jeep trails around Moab that would be a hoot on a V-Strom. There is (or used to be) a neat dirt road from I-70 almost all the way to Moab.
There are 4X4 and mountain bike guides that will tell you which roads are reasonably passable and which would require a trials bike and anti-gravity.
On a mountain bike trip a while back, we were able to get to some really neat places in a Toyota Corolla without undue trauma.
Also, go further west in Utah if you can. Everything all the way out to Zion National Park and beyond is absolutely stunning and amazing and...
2002 Suzuki DL1000 V-Strom, Dark Metallic Space Blue
1983 Suzuki GS850G, Cosmic Blue
2005 KLR685, Aztec Red - Turd II.2, the ReReTurdening