I don't have too much experience with that. I can't think of anything in the Show Low/Flag area. Problem there is more with private ranch land which all seems to be closed.
There is a dirt road from Winslow to Metero Crater south of I-40 that we wanted to take once. But I checked with a local and he told me the Hopis owned the land and it was closed.
We have always been able to ride on major BIA roads on the Navajo Rez. There is a nice section south of I-40 just west of the NM border and we have ridden quite a bit up around Chaco Canyon and Chinle.