I have been seeing and hearing misunderstandings of it. Here is one link to the piece and here are a few points:
1. Simulating the model, it is possible to generate something approximating the time series of U.S. employment changes, at least up through the 1990s.
2. This can be done by combining shocks to labor productivity, as measured (not calibrated to get the required result), combined with changes in search and matching profitability. No bizarre assumptions are required about the intertemporal substitutability of leisure, as the matching function does the work in this regard.
3. The results do not require sticky wages or AD shocks. Do not.
4. Mortensen and Pissarides do not explain "structural unemployment" in the traditional sense. Instead the model shows that one overall negative shock — to all sectors — can make good matches harder. In other words, unemployment can look like it is structural when it is not and in this regard the model can be viewed as a threat to traditional structural explanations. Furthermore unemployment can look like it is AD shock-induced even when it is not. Another way to read the model (my words, not theirs) is that it deconstructs the traditional distinction between cyclical and structural unemployment.
5. The model is built upon earlier theoretical contributions by Peter Diamond, but it is a mistake to speak of a unified Diamond-Mortensen-Pissarides model. I read Diamond's view as closer to Keynesianism and the Mortensen and Pissarides story as much more Schumpeterian. It is worth reading Paul Krugman's MIT-centric account of the prize, though I would stress the diversity of views among the winners.
6. Mortensen and Pissarides do not analyze monetary and fiscal policy in their basic model, but it is not obvious that such reflationary policies will solve the employment problem, which is defined in real terms, not nominal terms.
7. In the pure model, the Beveridge curve can have either slope and so we should not infer much from an observed Beveridge curve.
The Mortensen-Pissarides model is daring and subversive and it is impressive that the committee would award a prize for what is essentially a single paper. Yet I don't see too many bloggers trying to come to terms with it.