I used to think this way. I stopped thinking this way when I realized that there are discontinuities in consciousness even in bog-standard meat bodies -- about one a day at minimum, and possibly more since no one I'm aware of has conclusively established that subjective conscious experience is continuous. (It feels continuous, but your Star Trek transporter-clone would feel continuity as well -- and I certainly don't have a subjective record of every distinct microinstant.)

These are accompanied by changes in physical and neurological state as well (not as dramatic as complete disassembly or mind uploading, but nonzero), and I can't point to a threshold where a change in physical state necessitates subjective death. I can't even demonstrate that subjective death is a coherent concept. Since all the ways I can think of of getting around this require ascribing some pretty sketchy nonphysical properties to the organization of matter that makes up your body, I'm forced to assume in the absence of further evidence that there's nothing in particular that privileges one discontinuity in consciousness over another. Which is an existentially frightening idea, but what can one do about it?

(SMBC touched on this once, too.)