This is an interesting theory that has its roots in solipsism: the belief the world vanishes the moment you close your eyes. The fact of the matter is that it (and solipsism) just don't seem to be true, not even to solipsists. If you truly believed you were dying at every second, you'd act in all sorts of ways against your long term survival, but here you are posting on the internet.
Even if we take the position as true, it seems entirely reasonable to perceive the end of the "chain" of "yous" to be a tragic event.
> But who says that those living beings aren't just one big living being? Our brains tell us so.
Well, our brains, as well as the general evidence we see in the world around us. If we were one giant interconnected entity, dying would probably be very different. However, there's very little reason to assume that as a given - I can't talk to my dead relatives through any of my living ones.
I think this argument is perhaps more simply stated as "you were never a large part of the human race, so your death will not matter in a large way either" which is quite true, in fact. My counterpoint would be that the death of the individual rightly remains terrifying to that individual. If I were the human race, I would correctly not concern myself with the death of a few of my brain cells (and I do not, regarding my own real brain cells), but I would regard the death of the human race (myself) as truly important.
If anyone's interested, I've written about some of these kinds of arguments I've heard before: http://vincentwoo.com/2011/04/30/arguing-for-immortality/
If this doesn't work for you, then consider the following. We are all living beings, and sometimes some of us die, and sometimes new beings are born. But who says that those living beings aren't just one big living being? Our brains tell us so. But this view is only an artifact of the brain. If our brains were all interconnected more strongly, then I bet our brains would view us as one living being. So what you can take from this is that dying is just like losing a few brain cells. This is also something that you have been doing for years (in fact billions of years). So in this view, you never die until, of course, the universe ends.