UPDATE (JUNE 20, 2016) - Multiple readers have been commenting that U-Haul is currently NOT offering any type of discount. They have done this in the past when there has been high demand and no need to offer a discount. I wouldn't waste your time calling the number below given all the reports that no discounts are being given. I would still go online and check U-Haul, Budget, and Penske's price and compare them to make sure you get the best deal. In my past experience Budget has typically had the best price between the three companies.
U-Haul does not offer an "official" military discount. However, you can call the Sales Department at 800-468-4285 and they will often work out some type of discount. Their sales department is allowed to offer 10-15% off some reservations. Some of their representatives will give you that discount and some will not. Others have said that they will give you the discount but you must make the reservation at that moment. This 10-15% discount only applies if you are traveling one way (for example Phoenix to San Antonio). It typically doesn't work for "in-town" moves.
Sometimes this number (800-468-4285) will not directly connect to the sales department so always ask to be transferred to the sales department. They are the ONLY ones that can hook you up with a 10-15% discount. I used this a couple months ago when moving from West Phoenix to East Phoenix. It saved me about $15 last time on a $100 reservation.