First a comment: I am surprised at your stating that "It seems that physicalism and functionalism in philosophy of mind has fallen out of popularity and into much criticism". All of the philosophy of mind courses and lectures I have listened to state that functionalism is the most popular position in philosophy of mind. Can you provide a source?
Second: It should be noted that physicalism and functionalism are not the same thing. One can be a physicalist without being a functionalist (e.g. those who subscribe to type-identity or to eliminativism). In fact John Searle accuses functionalists of being closeted dualists (see these recorded lectures on philosophy of mind by him -- I forgot which lecture number he gives the argument for why functionalists are actually dualists).
Now for the answer. In what follows, I am describing arguments against functionalism as a physicalist theory of mind (i.e. ignoring other types of physicalism besides functionalism and ignoring Searle's argument that functionalism is in fact a form of dualism):
The main argument against functionalism is that it leaves something out, namely an explanation of subjective experience. Those who oppose functionalism maintain that it is not a complete description of the mind, that it fails to account for how or why subjective experience occurs. For functionalism to be a complete description of the mind, it would have to account for subjective experience, but in fact functionalist account cannot distinguish between a conscious being on one hand, and a being who has all of the functional and behavioral features of an intelligent human but doesn't have any subjective experience such as an intelligent AI robot or a Zombie that behaves outwardly like a normal human but in fact has no conscious experience (see the SEP article on Philosophical Zombies).
Thomas Nagel describes the problem in his paper "What's it like to be a bat" (Nagel, Thomas (1974). The Philosophical Review 83 (4): 435–450.):
We may call this the subjective character of experience. It is not captured by any of the familiar, recently devised reductive analyses of the mental, for all of them are logically compatible with its absence. It is not analyzable in terms of any explanatory system of functional states, or intentional states, since these could be ascribed to robots or automata that behaved like people though they experienced nothing.
David Chalmers proposed that this is the "Hard Problem of Consciousness" (Chalmers, David (1995). "Facing Up to the Problem of Consciousness". Journal of Consciousness Studies 2 (3): 200–219.):
It is undeniable that some organisms are subjects of experience. But the question of how it is that these systems are subjects of experience is perplexing. Why is it that when our cognitive systems engage in visual and auditory information-processing, we have visual or auditory experience: the quality of deep blue, the sensation of middle C? How can we explain why there is something it is like to entertain a mental image, or to experience an emotion? It is widely agreed that experience arises from a physical basis, but we have no good explanation of why and how it so arises. Why should physical processing give rise to a rich inner life at all? It seems objectively unreasonable that it should, and yet it does.
The argument against functionalism from knowledge
In my mind the most detailed argument for why functionalism fails to explain consciousness was provided by Frank Jackson (Jackson, Frank (1986). "What Mary Didn't Know". Journal of Philosophy 83: 291–295). In his original argument, Mary is a Neuroscientist and the sensation in question is color. I will illustrate it with a variation where Mary is a Dentist and the sensation in question is tooth pain, since it is more plausible in my opinion:
- Imagine a dentist called Mary, who is the top dentist in her field, she has aced every dentistry topic there ever was, can cure any patient who comes to her with a dental problem no matter how bad that patient's condition, and has studied every biological, neurological and chemical aspect of what a toothache is. In short she knows every single physical aspect and functional aspect of toothaches.
- However, Mary has been an avid tooth brusher ever since she was a kid, and she has never ever had a tooth ache in her life. Most would agree that because of this, she doesn't really know what a toothache is at all, since she has never had the subjective experience of a toothache. She knows all there is to know about toothaches, but she doesn't know what a toothache is, having never had the feeling herself.
- If Mary were to suddenly stop brushing her teeth and start eating copious amounts of chocolate, within a few months she will develop a serious toothache. The new experience she had constitutes additional knowledge of toothaches that she didn't have before she went through the subjective experience. It is argued that functionalism has no way of accounting for this additional knowledge that Mary gains from subjective experience. Functionalism incomplete, and there must a be a purely mental (or non-physical) aspect to things such as pain, color, love (i.e. qualia).
Possible functionalist responses
Daniel Dennett responds to this by claiming that qualia simply don't really exists in his paper "Quining Qualia" (Dennett, Daniel C., In "Consciousness in Modern Science" Anthony J. Marcel & E. Bisiach (eds.), [Book Chapter]. Oxford University Press (1988)) and later in his book "Consciousness Explained".
David Rosenthal, among others, proposes Higher Order Theories of consciousness to explain the existence of qualia/subjective experience within functionalism: Thoughts, beliefs, perceptions, etc..are 1st order mental (functional) states. Subjective experience are adequately described by 2nd order mental states. 2nd order functional states are functional states about other functional states.