We used to use cardboard boxes all the time out of JFK to the USVI...but have not tried since 9/11...Ours were good, brand new boxes, and we would strap them and then use the stretchy clear wrap outside....and they always made it just fine. Skycaps at JFK used to recognize us by our boxes....The TSA out of NY is our issue now, and the reason we stopped and switched to suitcases...
We have gone to thrift stores in the states and purchased luggage that we use just for the transporting stuff...and then either leave it at the destination or bring it back empty (since we get free luggage on AA) and use it again...no issue to transport a totally empty suitcase....and we are moving "stuff" for houses under construction or for us to use at our own house, so it's not clothing....I'd try for the plastic wrap at the airport if you have anything interesting in the suitcases, just to prevent pilferage...a constant problem for us now.