The most common justification of the method of moments is simply the law of large numbers, which would seem to make your suggestion of estimating $\mu_3$ by $\hat{\mu}_3$ "method of moments" (and I'd be inclined to call it MoM in any case).
However, a number of books and documents, such as this for example (and to some extent the wikipedia page on method of moments) imply that you take the lowest $k$ moments* and estimate the required quantities for given the probability model from that, as you imply by estimating $\mu_3$ from the first two moments.
*(where you need to estimate $k$ parameters to obtain the required quantity)
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Ultimately, I guess it comes down to "who defines what counts as method of moments?"
Do we look to Pearson? Do we look to the most common conventions? Do we accept any convenient definition? --- Any of those choices has problems, and benefits.
The interesting bit, to me, is whether one can always or almost always reparameterize a parametric family to characterize an estimation problem in EE as the solution to the moments of a (possibly bizarre) distribution function?
Clearly there are large classes of distribution for which method of moments would be useless.
For an obvious example, the mean of the Cauchy distribution is undefined.
Even when moments exist and are finite, there could be a large number of situations where the set of equations $f(\mathbf{\theta},\mathbf{y})=0$ has 0 solutions (think of some curve that never crosses the x-axis) or multiple solutions (one that crosses the axis repeatedly -- though multiple solutions aren't necessarily an insurmountable problem if you have a way to choose between them).
Of course, we also commonly see situations where a solution exists but doesn't lie in the parameter space (there may even be cases where there's never a solution in the parameter space, but I don't know of any -- it would be an interesting question to discover if some such cases exist).
I imagine there can be more complicated situations still, though I don't have any in mind at the moment.
Best Answer
Chapter 5: http://press.princeton.edu/titles/8434.html
Chapter 12: (Cameron&Trivedi textbook)
Chapter 15: (Greene's 7th edition)
This one discusses the general issues and is freely available: http://elsa.berkeley.edu/books/choice2.html
And I also recall that the handbook of econometrics chapter on simulation (40) is particularly readable.
Edited to remove extra links as only 2 are allowed.