Here's an eliminative materialist, Pat Churchland, from whom I get sort of a Miss Hathaway from the Beverly Hillbillies vibe. Keep a sharp eye for the key points where she is interrupted by Mr. Rogers music with pictures of traffic, and then later when she's overlaid with blurry students on campus:
Churchland here explains what "eliminative materialism" is supposed to eliminate. The phenomenon itself, i.e. consciousness, doesn't go away, but it gets causally explained, and the explanation will in turn enrich the concepts we use to describe consciousness. So, "folk psychology" talks about "the will," whereas Churchland doesn't think that there's going to be any single neural correlate to this term. Likewise "attention" we already understand to be several distinct systems, so an understanding of the causal basis might get us to be more precise in the way we talk about our inner lives.
For more detail, you can check out this article she wrote in 2005 called "A neurophilosophical slant on consciousness research:" http://philosophyfaculty.ucsd.edu/faculty/pschurchland/papers/progressinbrainsc05neuroslant.pdf
Here's an argument from there that's more contentious than anything in the video:
First, a common philosophical complaint is that any neurobiological theory of consciousness will always leave something out – something crucial. It will always leave out the feeling itself — the feeling of what it is like to be aware, to see blue, smell mint, and so on (Nagel, 1974; Chalmers, 1996). These are so-called qualia — the experiences themselves — and these are what are important about consciousness. Pursuing this point further, the philosopher may go on to conclude that no science can ever really explain qualia because it cannot demonstrate what it is like to see blue if you have never seen blue; consciousness is forever beyond the reach of scientific understanding.
What is the merit in this objection? It is lacking merit, for if you look closely, you will find that it rests on a misunderstanding. The argument presumes that if a conscious phenomenon, say smelling mint, were genuinely explained by a scientific theory, then a person who understood that theory should be caused to have that experience; e.g., should be caused to smell mint. Surely, however, the expectation is unwarranted. Why should anyone expect that understanding the theory must result in the production of the phenomenon the theory addresses? Consider an analogy. If a student really understands the nature of pregnancy by learning all there is to know about the causal nature of pregnancy, no one would expect the student to become pregnant thereby.
Like Chalmers's argument that accuses materialists of denying the raw fact of experienced consciousness (which Churchland doesn't), this argument is also a straw man: the discomfort with purely physical explanations doesn't arise because the theory itself needs to include the phenomena to be explained as part of the explanation (though Chalmers does believe that some non-physical elements have to play a part as basic explanatory principles of the universe); the discomfort is because the conceptual gap between the mental and the physical just seems fundamentally different to us than the gap between, say, the macroscopic phenomena of heat and the motion of molecules.
Churchland also makes the point that what science does is find correlations between phenomena, and if the correlations are tight enough, we call these identities, e.g. for all practical purposes heat is the motion of molecules. Science never explains WHY heat is correlated with that as opposed to something else; this is just a raw fact that we find, and hence every scientific theory of this sort provides some explanation in that it allows us to predict, and perhaps control, the phenomena, but this is not really ever sufficient for us to understand in a stronger sense what's going on. It's in the concept of the scientific method itself that prediction and and control is all we need; one of the philosophical questions here is whether that's really the case.