How do you request a 3D block of airspace to to freely maneuver your privately owned military jet?
As @jamessqf points out, once the aircraft is no longer owned by the government1 there wouldn't be any regulatory difference between you requesting air work and, say, a Gulfstream acceptance or post-maintenance flight that needs to fly around at odd altitudes and attitudes to check out various systems. That said...
Below FL180 you can fly VFR as you like. Of course you still have see-and-avoid responsibility, which might be helped by any onboard targeting radar on your military jet but then again might not be.
If you want to go higher (and have a smaller chance of turning someone's Cessna into a hood ornament) you need an IFR flight plan. Ideally you'd ask someone for permission to enter a Military Operations Area (example) and/or Air Traffic Control Assigned Airspace (which is the same concept as a MOA but exists in Class A airspace). FAA Order 7400.10 will tell you who the controlling agency and using agency are for each MOA; if you wanted you could try cold-calling Albuquerque ARTCC and asking if they'd take the Mt. Dora MOA hot for you, but the Air Force's 27th Special Operations Wing is the using agency and the line controller probably won't help you unless you run it by the military beforehand.
Assuming you can't get clearance into special-use airspace your only other option is to request a quadrant clearance in conjunction with a block altitude assignment.
A quadrant clearance (7110.65 4–1–1i) allows you to fly between two specified radials of a navaid and within two defined radii (DME arcs). Depending on the performance characteristics of your jet you might need a significant amount of distance, e.g. between the 10-mile radius and 40-mile radius, in order to get a large enough slice of airspace to work in.
A block altitude assignment (7110.65 4–5–7g) is more commonplace and allows you to operate anywhere between the two indicated altitudes. Again you'd probably want a good eight-to-ten thousand feet or more.
For both the vertical and lateral considerations you'd probably have the most luck over the Western states (ZLC, western ZMP, ZAB, etc). You could also try going offshore as long as you checked to make sure the warning areas weren't hot.

1Or operated on behalf of the government; I've seen a couple N-numbered British/European fighters operate using tactical callsigns as aggressor aircraft for military training exercises.