Well, on this, there isn't any doubt at all: since all the heart does is pump blood around, and since consciousness and identity reside in the brain and nowhere else, as far as we can tell, the person is the same person whether or not they have their original heart, lungs, liver, kidney, etc. And, since virtually all cells in all organs (except a few) turn over regularly, in a sense the heart you were born with isn't the same one you have now. Most likely, all the cells have been replaced naturally over the years.
If I recall correctly, though, the neurons in your central nervous system DON'T get replaced, so you have in some way the same brain (or lack thereof) with which you started out life.
Now- when we get to brain transplants, things are going to get a bit sticky..... (see jokes thread)