In more recent versions of Illustrator, one can simply Shift-Click to select more than a single artboard and move them.
Original answer posted in 2014
CS6 or older....
Unfortunately, Adobe hasn't seen a scenario where a user would want to move more than one artboard at a time. I'm not certain why this limitation exists. However, there's no way to manually move two or more artboards at the same time.
The only way to move more than one artboard simultaneously is via
Object > Artboards > Rearrange... which may or may not suit your needs. It allows you to organize the layout of artboards automatically. There are some limitations, such as no overlapping artboards and a strict adherence to a grid system, but I still find it useful at times.
In order for it to operate somewhat as expected, you need to "stack" or "order" the artboard list. For example, it will always assume the artboard created 4th is the 4th artboard if it appears as the 4th item in the Artboard Panel. You can adjust the (for lack of a better term) "stacking" order of the artboards by click-dragging artboards in the Artboard Panel into the order you want them. Then run the Rearrange command.