The term 'post-analytic philosophy' was first used in the mid-1980s. At that time it referred to those philosophers who were indebted to analytic philosophy but who nonetheless believed that they'd moved on from it (for whatever reasons).
The term 'post-analytic philosophy' seems, prima facie, odd. After all, how can philosophers be 'post'- or anti-analysis? Surely even most examples of post-analytic philosophy will contain analyses of sorts.
Thus the term must instead refer to the tradition of analytic philosophy. But which aspects of that tradition? Which particular philosophers? Did all analytic philosophers have an philosophical essence in common? And let's not forget that philosophical analysis occurred well before the analytic tradition got under way. (What is it that Hume, or Hobbes, or Aquinas, etc. were doing if it's not - at least partly - analysis?)
The above are all prima facie problems which, to some extent, subsides once the history and use of the term 'post-analytic philosophy' is studied.
However, it is indeed analysis that some philosophers seem to have a problem with. Or, rather, perhaps it's more accurate to say 'philosophical analysis' rather than the simple 'analysis'. This is obviously the case because the words 'philosophical analysis' are more particular than 'analysis' and it may/will contain assumptions as to what philosophical analysis actually is.
If we want to put meat on what post-analytic philosophers see to be the problem (or simply a problem) with analytic philosophy, it's best to consult late-20th century and contemporary American pragmatism. This school is itself seen as being part of the post-analytic movement (i.e., which isn't a determinate school).
Many would say that such American pragmatists have a problem with the very notion of objective truth. This is something they see as being an idée fixe throughout the history of philosophy. This, indeed, was no less the case when it came to 20th and 21st century analytic philosophy.
My prima facie personal objection to this is that I've hardly read a single analytic philosopher mention - or use - the words “objective truth”. (I have read, however, Peter van Inwagen's 'Objectivity'.) Then again, it can easily be argued that a philosopher needn't use the actual words “objective truth” in order for him to be committed to the notion of objective truth. In other words, perhaps he simply calls it by another name.1
In any case, the position that objective truth doesn't exist, or that it's not a worthy aim in philosophy, goes alongside a stress on the contingency of cognitive activity, the importance of convention and utility and, indeed, the idea that human (or social) progress can never be ignored – not even in philosophy. Nonetheless, here again I don't see how there's an automatic or prior problem with accepting all this and still engaging in analytic philosophy (or in philosophical analysis). In very basic terms, for example, one could offer a philosophical analysis of philosophical analysis (or some part thereof); and then, as a result, see philosophical problems with such philosophical analysis. Despite that, such a philosopher would still be in the domain of analytic philosophy (or of philosophical analysis).
Strangely enough, Rorty seems to agree with this position. Or, at the least, he says something similar. In an interview conducted by Wayne Hudson and Win van Reijen, Rorty states:
"I think that analytic philosophy can keep its highly professional methods, the insistence on detail and mechanics, and just drop its transcendental project. I'm not out to criticize analytic philosophy as a style. It's a good style. I think the years of superprofessionalism were beneficial." 
I said the position is “similar” to the one advanced by Rorty. It's similar in the sense that an analytic philosopher needn't “drop [his] transcendental project”. That is, an analytic philosopher may be fully aware of Rorty's positions/arguments (or the general positions of post-analytic philosophers) and still be committed to the transcendental project. (Of course we'd need to know what Rorty means by the words “transcendental project”.)
Philosophy Must be Political?
It seems that the position of many post-analytic philosophers is primarily political - or at least primarily social – in nature. Putnam (1985), for example, has said that analytic philosophy has “come to the end of its own project—the dead end”. That can be taken to mean that philosophy should connect itself more thoroughly with other academic disciplines. Or, more broadly, that analytic philosophy should connect itself with culture or society as a whole.
The problem with this is that, on and off, analytic philosophy has already connected itself to many other disciplines. Admittedly, that's been more the case since the 1980s and the rise of cognitive science. However, to give just a couple of examples, the logical positivists connected themselves to science (or at least to physics). And, to give one other example, philosophers in the 19th century connected themselves to logic and mathematics and, again, to science. This non-ostentatious “interdisciplinary” nature of philosophy has been true, in fact, throughout the history of philosophy.
One can also say that philosophy can connect itself to other disciplines - and even culture as a whole - and still remain analytic philosophy. Philosophers can still practice philosophical analysis. (This, again, raises the question as to what analytic philosophy - or philosophical analysis - actually means in either narrow or broad terms.)
A philosopher may also ask why he should connect himself to other disciplines, never mind to something as vague or as broad as culture. In other words, a philosopher must have philosophical reasons as to why this would be a good thing; just as a philosopher must have philosophical reasons as to why it's a bad thing. That means that there'll be philosophical angles to this very debate; though, it can be added, those angles needn't always be philosophical in nature.
Another slant on this philosophy-society binary opposition is that analytic philosophy is too professional and therefore too narrow. In other words, analytic philosophers are over-concerned with very tiny, narrow and specialised problems which have almost zero connection to society as a whole.
More technically and philosophically, it can also be argued that certain central commitments and assumptions of analytic philosophy have been shown to be indefensible. (Hence Putnam's own words quoted earlier.)
All disciplines can be said to concerned with narrow or specialised issues or concerns. Yet this is an accusation more often than not aimed more at philosophy than any other academic subject.
I would say that Richard Rorty appears to be talking about analytic philosophy as it was in the past (say, the 1950s to the 1970s), not as it is today or as it's been since, say, the 1980s.
Take the view that analytic philosophy has as its primary aim a form of knowledge which, as it were, grounds all other forms of knowledge. This is odd. It's true that much traditional philosophy has placed various philosophical domains as what used to be called First Philosophy. (It was once metaphysics, then epistemology, then language, then mind...) However, in the 20th century this has been far from the case. Indeed philosophers - throughout the 20th century - have argued against the nature of a first philosophy. Take naturalists for a start (e.g., the logical positivists then Quine): they placed science or physics in the role of first philosophy. (Although such naturalists saw physics as being primary, that isn't in itself a commitment to also seeing it as some kind of first philosophy.)
It can be said that just as it can be said that Rorty's post-philosophy is a philosophical position, so too is the Wittgensteinian attempt to “dissolve” and then disregard philosophical problems (if not philosophy itself). This position can be said to be held by Putnam and McDowell, as well as by Rorty. And a more specific example of this would be the “problem” of how mind and language are connected to the world.2
In any case, there's just as strong argument for saying that Rorty's later position was more a case of post-philosophy than post-analytic philosophy. In other words, like Heidegger and Derrida, he had a problem with the whole damn show that is philosophy. And, here again, it can be argued that Rorty's position was more political or social, rather than strictly philosophical. Though, of course, a position that rejects philosophy in toto can't help being philosophical – in some or many ways – itself; as Rorty would have no doubt happily admitted. (Jacques Derrida did admit this.)
1 For example, if you say that an argument or a single statement is “warranted and therefore assertible”, is that a case of being wedded to the notion of objective truth? Or is the notion of warranted assertibility a different species entirely?
2 I put the word “problem” in scare quotes because the very stance of seeing such problems as problems means - according to Rorty, Derrida, Heidegger, etc. - that we've fallen prey to particular philosophical “style of thinking”. But that too would be a philosophical position!
Mendieta, Eduardo (2005) Take Care of Freedom and Truth Will Take Care of Itself: Interviews with Richard Rorty (p. 23).
Putnam, Hilary (1985) ‘After Empiricism’ in Rajchman and West's Post-analytic Philosophy.