I will sometimes use a derived sketch for this. I'll past the sketch on one of the main planes, then I select it and the plan I want to put the sketch on, then go to the insert pulldown where you'll find "Insert Derived Sketch". This forces an exact copy of the original sketch to be placed on the new plane. Because it doesn't allow you to make any changes to the derived sketch it moves the sketch around as a single entity. Once I'm done placing it with relations I usually fix all the entities, then exit the sketch, underive it from the right- click menu, and finally delete the original.
I usually only do this for large sketches. I've used the Ctrl key trick that fcsuper mentioned: select entities, hold Ctrl, start to drag, then while still holding the left mouse button release the Ctrl key to change the "copy" to a "move". The problem I sometimes run into with this is if the sketch has some short line segments I've had it shift some of the end points around.
I like the idea of trying to make it a block before you move it around. I might try that first next time instead of the derived sketch method.
Rory