You might think, by his preceding paragraph, that this applies to all of his "trickle-down theorists". It emphatically does not apply to Penrose or John Searle -- two of the three he names explicitly. (I'm not even sure that it applies to Descartes.)
So, let me just summarize those two, for people who aren't so familiar with their writings. Penrose is a physicist who thinks it peculiar that we can mathematically prove, for any computer, here is a true fact which that computer cannot prove. He thinks this is peculiar because if our understanding were algorithmic, then that would imply that we could eventually write down its axioms as a formal system, go through the Godel proof line by line, and prove, "here is a true fact which my understanding cannot prove," thereby using our understanding to prove that fact, thus constructing a contradiction. His conclusion is that understanding can't be algorithmic. (Dennett of course disagrees.) Nonetheless, he believes in science. He thinks that we need to understand existing science (quantum mechanics and perhaps gravity) better to comprehend consciousness, but he certainly seems to believe that science will eventually comprehend it.
Searle is a philosopher who finds it peculiar that functionalists (like Dennett) still seem to quietly accept the Cartesian counting of substances. That is, they seem to accept that consciousness is somehow a 'thing' which is distinct from the stuff of the world, and therefore they must argue that such 'things' don't really exist. Searle wants to say, first and foremost, that we really are conscious when we wake up in the morning -- all that touchy-feely crap that you experience until you go to bed at night isn't some sort of illusion, it really is a part of the world and does affect the world, etc. Again, he views it as a mistake to think that it's somehow a 'second substance' which cannot be reconciled with 'material substance' -- they're both parts of the physical world and whether you treat them separately says more about you than it does about the world.
He also wants to note that it (consciousness) is not a computation. This is for a very simple reason: whether a box is performing a computation is observer-relative, given Turing's definition of a computer as a symbol-shuffler: an observer who doesn't interpret the symbols 'properly' doesn't see it as 'computing' anything. But whether you are conscious is presumably observer-independent; there is no perspective that I can switch to by which I might somehow obviate your consciousness and thereby render you incapable of feeling, for example, pain. Many of Searle's articles contain the attitude of: I'm going to clear away the philosophical problems for once and for all and then the scientists can do the hard part of figuring out what neuronal events correlate with consciousness and all of that stuff, so that we can understand consciousness. In that sense he certainly believes that science will eventually understand consciousness.
These two might not even agree that "a thinking thing cannot be constructed out of Turing's building blocks." If Dennett's point is "sorta-thinking" as a model for thinking, then they might agree that a program can sorta-think. Penrose is merely skeptical that sorta-thinking can climb as high as understanding, while Searle is skeptical that sorta-thinking can rise to the observer-independent nature that we consider a fundamental part of our everyday experience.