I hate to be sarcastic, but Eugene Fama built quite a career for himself by running linear regressions. That said, the website for EMM http://public.econ.duke.edu/~get/emm.html indicates that it should be used for otherwise intractable maximum likelihood problems. So I would say, instead of fad based methodology, the structure of the problem dictates admissible methodology. MCMC would probably subsume EMM for pure data intensive parameter estimation. But EMM should give better diagnostics because it appears to be based on identifying restrictions on a structural model.

I only use MCMC these days, but partly because it is very easy now with Stan software. See Preprint Building and Testing Yield Curve Generators for P&C Insurance

for example. You might need to create a free account to get it.