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.
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.