They do not stamp your passport at each port. In fact, after you show it to the cruise rep (ie Carnival) at check-in you will not need your passport for the remainder of the trip. Just keep it in your cabin safe or luggage.
I have heard of some people getting stamps at some ports, but I am not sure if this is true.
NEXT: Royal Caribbean Navigator of the Seas 3/11/2012 Western Caribbean
Royal Caribbean Majesty of the Seas 12/4/2009 Bahamas
Carnival Valor 3/9/2008 Eastern Caribbean
Royal Caribbean Voyager of the Seas 6/24/2006 Western Mediterranean
Royal Caribbean Freedom of the Seas 5/29/2006
Carnival Conquest 6/20/2004 Western Caribbean
Royal Caribbean Sovereign of the Seas 12/20/2002 Bahamas