The best (so far only imaginary) device to study the conscious mind is a teleporter. It is a device that can take a complete scan of your body, including the state of your brain. The scanning is destructive, but not to worry—another teleporter can reconstruct you exactly how you've been at the moment of the scan. I will not go into all the technical difficulties of doing that. Some people will claim that in order to reconstruct the mind one would have to reproduce the quantum state of every particle in the brain. Since the full quantum state (wave function) is not measurable, we would be in trouble. However, we'll assume that it's enough to know the positions and velocities of all the particles in our body to some reasonable approximation in order to reconstruct the mind. After all, our mind seems to be pretty insensitive to all the thermal fluctuations that constantly rearrange the particles in our brains so why should it be sensitive to small measurement inaccuracies.
Anyway, we are not actually building a teleporter; we are only going to use it for gedanken experiments. First of all, there is a point of view that says that destructive scanning is a form of death, and that your freshly reconstructed double is not a continuation of you. (In fact, for some reason, I don't think I would like to undergo any of the experiments I will describe.) However, if the teleportation is successful, your double will have full memory of everything that happened to you, up to the point when you were scanned. If somebody asks your double if he (or she) is the one, he'll answer to his best knowledge that indeed, he is you, and that he didn't experience any discontinuity in his consciousness. All he knows is that one moment he was in the transmitting teleporter and the next moment he was in the receiving teleporter. Other than that, nothing unusual happened. There is no reason for us to doubt his account.
Somebody might argue that there can't be the same consciousness in your double as in your original, because there was no physical continuity. But people survive drastic discontinuities and nobody ever doubts that they are the same person as before. You may get knocked out unconscious, go into a coma for months, undergo a surgery under deep anesthesia, and you will still consider yourself the same person as before. If all these traumatic discontinuities don't make us doubt our identity, why should we worry about teleportation?
Well, there is a difference. We have assumed that the scanning is destructive, but what if it isn't? What if one of you comes out of the transmitter and another one from the receiver? Which one is the real you? Most people would say that the guy in the transmitter is the real you. But ask the transmitted one, who came out of the receiver, and he will swear that he's the one. And if you believed him before, when the scanning was destructive, why would you doubt him now? You have to accept that both are a continuation of you. There is no objective way of disproving it and, subjectively, they both are totally convinced of their own authenticity. By the way, this argument fits nicely with our earlier description of the many-universe interpretation of Quantum Mechanics. There too we had to accept the equivalence of all the "branching observers."
The purpose of this warm-up exercise was to convince ourselves that the objective and the subjective perceptions of time don't have to match. The subjective time is the function of our brain's state, the memories and the expectations. But we can come up with (so far imaginary) situations where our sense of linearity and continuity of time can be fooled. So the mere fact that we perceive time as a straight arrow doesn't prove anything.