There are several national grasslands in North Dakota and South Dakota; I'll be visiting four.
Cedar
River National Grassland. This grassland is just over the state
line from South Dakota; it's not large and is checkerboarded more than the
others. You'll need to know how to find it as it's not likely you'll
ever just stumble on it while driving. I believe all of it (or nearly
all) is within the Standing Rock reservation.
Sheyenne
National Grassland. This is in southeast North Dakota and has an
impressive mix of trees and dunes and tallgrass prairie. The
North Country
Trail passes through the grassland.
Grand
River National Grassland. This grassland is not far south of Cedar
River Grassland; Grand River is far larger.
The Blacktail Trail looks new and takes you on a roughly ten mile loop. They've done a nice job with it.
There are a few unpaved roads through the grasslands, and some are just fine for normal vehicles (at least if it hasn't rained).
Fort
Pierre National Grassland is just south of the state capital city,
Pierre. For wide open vistas, this is a good one.