Alternatives to the Baron-Kenny approach to modeling mediation I'm about to open the door to a very thorny issue in the social sciences.  How does one correctly model and test hypotheses about mediating variables using observational data?
I'm familiar with the Baron-Kenny approach to mediation (see previous answer here), and also with structural equation modeling.  However, I've heard both approaches disparaged by more quantitative-minded social scientists than myself -- especially when one is using observational rather than experimental data.
So, let's say that I'm trying to resolve the following: 
Y is a behavioral outcome. Both X and Z are observed characteristics of subjects that cannot be manipulated by an experiment. X is an attitude (something that can be changed over the long term) and Z is an unchangeable characteristic such as age, race, etc. 
I hypothesize that X is a mediating variable, thus Z affects Y through the pathway of X.  While it's reasonable to suggest that Z is in some way correlated with X, my theory argues that it has no impact on Y (other than through **X).
How would one best test these hypotheses using best practices in current research?
 A: Baron and Kenny are indeed outdated, though that does not make them wrong in all cases.  
The concerns divide into broadly statistical limitations and assumptions which are discussed in the reference your reviewer suggests and in the literature alluded to by @PeterFlom, and broadly non-statistical concerns about the definition and causal identification of mediation effects.
Following this order it might be helpful to start your reading with MacKinnon et al.'s 2007 review, or with the reviewers suggested reading.  Then move on to Imai et al. 2010a (or Imai et al. 2010b) These last papers are dense, but repay study.  that should bring you more or less up to speed on how mediation analysis is being thought of lately.
A: This is more a discussion of concerns I have firstly with the approach of Baron and Kenny (which has some bearing on your question), and with a number of more recent papers (I haven't seen them all, so my comments may not apply to everything). It may also relate to the 2011 paper you mention, which I have only had the chance to skim through just now.
From what I've seen, the idea of measuring/establishing mediation mostly seems to suffer a basic problem* that I haven't seen adequately dealt with. (I've just taken a fairly quick look at the 2011 paper you mention, so maybe I missed something. The example in figure 2 of the paper seems to be related, which is encouraging in the sense that at least some possibilities are being mentioned in some parts of the literature now.)
* The first time I ever heard of mediation and read a copy of Baron & Kenny, I saw this would be an issue. It seemed to be a surprise to most people I mentioned it to. 
The problem is this - to establish that $M$ actually mediates $X\to Y$ (at least partly), as below:
$\quad $ 
(the dashed arrow indicates a reduced level of relationship), it is necessary (for example) to rule out all feasible explanations like these in place of the second diagram:
$\quad $ 
(The grey variables might be latent, unknown, unaccounted for - or in some other sense 'hidden' from the model, or the researcher, or perhaps even anyone. There may also be some direct relationship between $X$ and $Y$ as well, it makes no real difference to this issue.)
Many papers (at least many of the ones I have seen) which deal with mediation, when they follow the recipe that is supposed to establish whether mediation happens, immediately start saying things along the lines of "$M$ mediates $X\to Y$" and discuss the size of this effect, when unless they have eliminated essentially any possibility of such hidden variables actually driving the relationship in any number of arrangements and variations, they really haven't established such a thing at all, and any measures of the size of the mediation effect rely heavily on those other possibilities not being present, at least not to any substantive degree.

An additional issue is that methods such as regression don't "put heads on arrows". To do so with such methods requires careful experiments, or very careful argument; if both are missing, generally speaking all really we have is correlation, and correlation is not the same thing as causation.

I am hoping one of the very good quality statisticians here will be able to school me on why my concerns are mostly unfounded; such an education would be most welcome.
A: Here are some places to look. I'd especially recommend the work by Kosuke Imai and colleagues.
Bullock, John G., and Shang E. Ha. 2011. Mediation Analysis is Harder Than it Looks. In Cambridge Handbook of Experimental Political Science, ed. James N. Druckman, Donald P. Green, James H. Kuklinski, and Arthur Lupia. New York: Cambridge University Press.
Bullock, John G., Donald P. Green, and Shang E. Ha. 2010. Yes, But What’s the Mechanism? (Don’t Expect an Easy Answer). Journal of Personality and Social Psychology 98 (April): 550-58.
Fiedler, Klaus, Malte Schott, and Thorsten Meiser. 2011. “What Mediation Analysis Can (Not) Do.” Journal of Experimental Social Psychology. doi:10.1016/j.jesp.2011.05.007. http://dx.doi.org/10.1016/j.jesp.2011.05.007.
Green, Donald P., Shang E. Ha, and John G. Bullock. 2009. “Enough Already about `Black Box’ Experiments: Studying Mediation is More Difficult than Most Scholars Suppose.” The ANNALS of the American Academy of Political and Social Science 628 (March) (January): 200-208.
Imai, Kosuke, Luke Keele, Dustin Tingley, and Teppei Yamamoto. 2011. “Unpacking the Black Box: Learning about Causal Mechanisms from Experimental and Observational Studies.” American Political Science Review 105 (4) (November 10): 765-789. http://imai.princeton.edu/talk/files/ISM10.pdf.
Imai, Kosuke, and Teppei Yamamoto. 2011. "Identification and Sensitivity Analysis for Multiple Causal Mechanisms: Revisiting Evidence from Framing Experiments."
MacKinnon, David P., Amanda J. Fairchild, and Matthew S. Fritz. 2007. “Mediation Analysis.” Annual review of psychology 58 (January): 593-614. doi:10.1146/annurev.psych.58.110405.085542. http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2819368&tool=pmcentrez&rendertype=abstract.
Spencer, Steven J., Mark P. Zanna, and Geoffrey T Fong. 2005. “Establishing a Causal Chain: Why Experiments Are Often More Effective than Mediational Analyses in Examining Psychological Processes.” Journal of Personality and Social Psychology 89 (6) (December): 845-51. doi:10.1037/0022-3514.89.6.845. http://www.ncbi.nlm.nih.gov/pubmed/16393019.
A: Baron and Kenny is distinctly old fashioned these days. They see mediation as a "yes-no" "present-absent" quality; more recent approaches (lots of work by MacKinnon and others) treats it as a continuum. This makes more sense to me. 
A: I agree with the above answer, and I would like to add more information in a form of a succinct summary.
Baron and Kenny's (1986) method of testing mediation has been extensively applied, but there are many papers discussing severe limitations of this approach, which broadly include:
1) Not directly testing the significance of an indirect effect
2) Low statistical power
3) Inability to accommodate models with inconsistent mediation
*Note: see Memon, Cheah, Ramayah, Ting, and Chuah (2018) for an overview.
Considering these limitations, a new typology of mediation was developed by Zhao, Lynch and Chen (2010). As of Oct 2019, it has over 5,000 citations, so it is gaining greater popularity.
As a brief summary, and taking a three-variable causal model as an example, thee types of mediation exist.
Complementary mediation: Mediated effect (a x b) and direct effect (c) both exist and point at the same direction.
Competitive mediation: Mediated effect (a x b) and direct effect (c) both exist and point in opposite directions.
Indirect-only mediation: Mediated effect (a x b) exists, but no direct effect (c).
Further, two non-mediation types were proposed:
Direct-only non-mediation: Direct effect (c) exists, but no indirect effect.
No-effect non-mediation: Nether direct effect (c), nor indirect effect exists.
References
Memon, M. A., Cheah, J., Ramayah, T., Ting, H., & Chuah, F. (2018). Mediation Analysis Issues and Recommendations. Journal of Applied Structural Equation Modeling, 2(1), 1-9.
Zhao, X., Lynch Jr, J. G., & Chen, Q. (2010). Reconsidering Baron and Kenny: Myths and truths about mediation analysis. Journal of Consumer Research, 37(2), 197-206.
