The questions of personal identity have been keeping philosophers happy for centuries. It does seem to me that we need to answer this question before we can speculate on which part of us might survive death. Lots more questions are raised by the teleportation thought experiment, but I was merely suggesting that if you don't mind being teleported, then you can have no logical problem with an afterlife even without a soul.
For the rest of us, maybe it is easier to ask what we are not. We are not just the particular atoms that make up our brains. We know this because these atoms are replaced by different ones quite regularly (as radioactive material injected into our system has demonstrated). Neither are we wholly detached from our bodies as shown by many neurological experiments. They show we need a body to be conscious and interact with the world. This means the Bible is probably right to say that the afterlife will have to involve our having new bodies. John Polkinghorne suggested that we are a pattern that presently just happens to be made out of material atoms but could, in theory, also be contained in another medium. This is quite close to materialism and he also goes on to say that immortality might be being remembered in the mind of God.
On the other side of the equation is the brute fact of freewill. I have previously shown how freewill is a necessary quality of consciousness. If we cannot will then we cannot do anything. And if we are not doing anything then we don't exist. This is what some people admit when they say consciousness is an illusion. In other words, they go even further than Descartes when they say that they even doubt their own existence. This leaves me feeling that materialism, that has no room for freewill, is inadequate as an explanation for mind.
I think that Thomas Aquinas gives us the best answer when he describes the immortal soul as growing with the body and giving it the capacity for freewill. The soul is an enabler and catalyst that needs the body in order to fulfill its potential. Deprived of sense experience and the machinery of thought it can do very little. So perhaps we should step back from Cartesian dualism and return to Aquinas's more subtle analysis. It is worth noting that Aquinas also rooted the emotions and desires firmly with the body and not the soul. Thus he would have been unperturbed by research that shows the brain doing a lot of what Descartes might have reserved for the soul. Also, he accepted that the body/soul relationship was a two way process and that the soul is formed from its experience in our earthly body. It grows and adapts as we grow and adapt. When we die, it is our ability to will that survives and which is given a new body. The mystery that Aquinas leaves us with is how to connect the body and soul. This mystery remains and I welcome suggestions!