I posted earlier on this topic or rather tried to post but hit the wrong button somehow and lost it. But anyway, if you look up the ADV riders website there are multiple blogs on the GDMBT. I think you can do the entire thing on a dirt bike and from what I have read in the past you can do almost all of it in a 4x4. There have been both bicycle and dirt bike groups that have used 4x4 sag wagons.
The entire route has been sliced, diced, detailed, way-pointed, video'd, photographed and otherwise documented in more detail than most of us could ever absorb. If you want a slightly more spontaneous trip, (real adventure perhaps?) just ignore the detail but it does give a pretty good idea of where you can actually go on dirt roads as a starting point. In NM the route ignores some very cool places that are not too far off the divide for a vehicle. I would stop in Mogollon, not far from Silver City, perhaps spend a day looking for wolves in the Gila, maybe go over Mt. Withington which is a very pretty dirt drive, perhaps detour through Winston and Chloride and over to Truth or Consequences for the hot springs and general funkiness. The El Malpais national monument is really interesting, the Zuni's are almost completely empty and can be crossed by dirt road, ending up near Gallup, (100 miles west of the divide) you can drive over the Mt. Taylor massif from Grants on dirt, down towards Cabezon (I think this is on the GDMBT), Cabezon has an easy hike/scramble on the NE side that is well worth doing for the views, eat at El Brunos in Cuba, do hit the ruins in this part of the world, especially Chaco canyon though you could spend weeks exploring the other ruins in the area.
Then a bit of Apache reservation and you're in Colorado. Depending on your interests, or level of disconnectedness from the reality most of us occupy, there's also Dulce, NM, Famous for some sort of alien life form/force/whatever.... http://www.reptoids.com/Vault/ArticleCl ... istory.htm
Sounds like an great opportunity for a fabulous trip, keep us posted!