The on board (any cruise line) satellite system for calls, WMS (aka cellularatsea) does not work with pre-paid phones so the burners are no good until you are in some ports. Many islands have their own mobile phone companies some of which work on the same transmission system as European countries and many cheap disposable phones just are not compatible. If you really want to go this route do very good research on the type of pre-paid phone and the calling cards for it you may need in each port.
OR save your money and check with your service provider about vacation international plans. Though this doesn't help with accidental transmissions or data reception on board, most "contract" plan phones are more compatible for in port use. Once on board, turn off your data, drop your quick dial widget off your start page, turn on wifi only use, and be careful about what setting your apps are using (such as auto-update which could override a manual data off option).
If in doubt just turn off your phone and leave it in the cabin's safe, dig out an MP3 player or e-reader for music or reading pleasures and pick up a cheap digital camera (I've seen waterproof ones with 10 MP at Staples for under $30) for taking pictures.