Questions like these are why I have something of a problem with most modern science fiction that popularizes various ideas of brain storage and transfer, "resurrection" via clone duplicates, etc, because the authors often have the characters in these stories treat these hypothetical technologies as if there's no concern or question of what consciousness is or whether their personal point of presence is preserved. Some authors (like Peter Hamilton) even go so far as to have some characters chortle at a few cranks with "old fashioned views" - those old fashioned views are in fact, the objection people /today/ would have to "transport" and mind-transfer technologies that were not guaranteed to move one's personal point of view in space-time to a new location / new body.
Simply put, I think that it's bollocks that most people (in a fictional setting where such technologies existed) would blithely go along with them even if all the atheists managed to stamp out every trace of religion and all people were nice, sane rational materialists. I realize SF authors are stuck here, because we don't /have/ great answers to these questions right now, in real life. The problem is that it's one thing to invoke Handwavium in relation to making a starship go faster than light. It's another thing to wave one's hands when it comes to people accepting an everyday transportation and medical techniques that just so happen to destroy them and leave clones behind to carry on.
In order to create some suspension of disbelief myself when reading such stories, I tend to muck about with a hypothetical idea of consciousness as a non-localized phenemonon, somewhat related to some early, proto-science theoroes about possible quantum mechanical aspects of consciousness. In one of my models, it's not so much that there is a "soul" behind the operation of the mind; as what we could call the soul is a more universal phenomenon and any particular instance of mind is just one particular pattern being evoked out of this conscious ether (ha! Ether! It sounds so 19th century! Mmm, funny hats...).
So when you're taken apart and reassembled on the other end of the transporter pad, "you" are not destroyed, because "you" never existed in the first place as a single entity. The physical brain that tugged on a string of universal consciousness is being rebult in another place, and because its structure is absolutely identical, the instance of consciousness that it evokes is effectively the same one as before. From "your" perspective, you black out and then wake up again, and yes, it really is "you", inasmuch as "you" actually exists in the first place. There, personal continuity preserved
(Interestingly, in this little nugget, if for example a teleporter malfunctioned and left "you" on the pad as well as threw you to the other desination, both yous are entirely real and the same point of prescence in space-time. As far as the quantum ether (ha! ha! Funny hats!) is concerned, there's now two outlets for presenting itself in one particular form or pattern, where before, there was only one.)