I can definitely see why reductionism would appeal to an educated public. We learn about the periodic table; we are able to break sentences down to words, down to letters; to break an executable down to binary code, to machine instruction, to electrical currents flowing through semiconductors. Why shouldn’t we be able to do the same with conscious thoughts? It is certainly an appealing thought process.
As I understand it, reductionism started to fall out of favor because of the rise of quantum mechanics and chaos theory, where we have a lot of weird phenomena which cannot be explained by reducing the particles down to the sub-sub atomic (or rather they are better explained by describing the interactions directly).
Also, nothing about the idea of the mind only being made up of physical processes means things have to be deterministic.
There are a whole lot more physical processes going on in our bodies then just neural activity. And my best guess is that is exactly where reductionism fails. It is possible that neural activity is a necessary but not sufficient condition for consciousness. It is also possible that we are looking in the wrong direction, that consciousness arises via interactions with the world. In either case (of which I find the former quite convincing) we will never be able to describe the mind by just looking at neural activity.
I am actually of the opinion that cognitive scientists are doing an excellent job describing the mind with our current theoretical models which excludes the tough questions of consciousness.