"Cycling" is the act of establishing a bacterial population suitable for handling your nutrient load. The bacteria generally live on rock and sand. Transferring rock and sand will preserve most of the bacterial population, which would achieve your goal.
That said, transferring sand can be problematic. Many times, older sand will be full of decaying organic matter. It doesn't cause a problem in your tank because it's locked away under the sandbed. But, if you transfer the sand, you'll be stirring it up and releasing that organic material back into the water column. This can easily kill a tank if the sand is polluted enough, but if the sand is clean, it's often not a problem to re-use the sand.
We can't tell where your tank is on that spectrum, you'll have to look and decide - or just discard the sand to be safe. On the other hand, transferring the rock is usually very safe, although you may stir up some gunk just doing that. Regardless of what you do, have enough saltwater on hand to do one or two large water changes within the first day or two of the new tank's life, just in case.
The best way to handle this would probably be to set up the new tank with completely new sand, seed it with a few handfuls of the old sand, and let the new system run for a week or two in order for everything to stabilize and the bacteria to populate the sandbed. Then, move your rock and livestock into the new system. If you're putting the new tank in the same location as the old one, or you're re-using all the equipment, this might not be possible. In that case, you could pre-cure the new sand in a rubbermaid tote with a heater and small powerhead, or just go with all new sand (plus a handful of old to seed it) and hope for the best.
At any rate, I wouldn't transfer the old sand unless you were very confident that it was clean.