Okay, I'll tell you how I do it. Some people disagree with my method, but it works for me every time.
ctrl-a for select all. In windows, hold alt and drag-copy with shift above or below the graphic. Distance doesn't matter, because the distance you use will stay in the dialogue box later on. Make every element in the design the same color with no stroke.
I use the 'merge' command on the pathfinder toolbar, commonly known as 'weld'. When you've got one graphic that contains all the elements of the design, offset it however far you choose.
Once your satisfied with the results hereto, I use ctrl-shift-m to move. If you haven't moved anything since copying your stuff, the exact distance you moved it all before will still be in the box. Apply the opposite sign to move it the other way, and it'll fall right on top of the original.
Apply the CutContour.
As I said before, some don't like this way, but it works. (for me)