The Bureau of Land Management runs much of the open West, and like the national forests, it is mostly dog-friendly. Its national monuments and conservation areas are some of the best dog country there is.
Every unit below welcomes leashed dogs, checked against the BLM's own pages. Expect red rock, desert, and canyon country, often with hardly another soul around.
A 6-foot leash is required in developed areas, and out on the open land your dog needs to be under control. Carry more water than you think you need.
BLM land is where the West opens up for a dog. Pair a monument with a nearby national forest and you could hike for a week.
A 6-foot leash is required in developed areas like campgrounds and trailheads. On the open backcountry there is often no leash law, but your dog must stay under control, and local offices can set stricter rules.