I asked Dan Everett for his advice, and he replies:
One can do field research from any speciality of linguistics, hyphenated or not.
One needs to have the interest in it, some training, and some talent for crosscultural understanding and learning other languages.
For training, I would suggest Sakel and Everett (2012).
Find a part of the world you are interested in, try to meet people from there, especially local linguists, and read on the languages there. Money for research will be hard at first, at least it often is, so try to get local university grants. Go on a shoestring budget and get some good data based on your own interests (driven by knowledge of typology, theory, etc). Once you have returned and written up something, use this to develop a larger project for more significant funding.
This doesn't exhaust advice and I am sure Rich will have ideas too. But there is no one way to do it.
I also asked Rich Rhodes, and he replies:
I guess I look at this from an odd angle. I think a reasonably talented linguist with MA level training is enough preparation to go to the field. I'd like to suggest that real success depends on stuff they don't teach: the willingness to get personally involved in the community you study, patience, humility -- especially humility -- the ability to think outside the box (as the cliché goes), an eye (and ear) for serendipity, an unwavering respect for the data, and the rare ability to be obsessed with getting the details right without losing sight of the big picture.
I'm sorry if this sounds a little zen, but I think most of what makes for the best fieldworkers isn't really taught and may not even be teachable.
I suppose I should add a word about what I mean by reasonably talented. You have to have a good ear. Period. You have to be able to do basic morphemic analysis in your sleep. And, possibly the hardest thing to come by in this day and age, you have to be able to do pre-theoretical syntax and semantics.
You will, I guarantee it, run across phenomena that don't fit anyone's theory. The worst possible thing to do is to gloss over something because you have no theoretical/conceptual box to put it in. You have to have the tools to explore it even if you can't explain it.
For the record, I agree with both of them, and appreciate their responses. And I'd add that fieldwork is not cheap, and one normally spends at least as much time and effort on getting the wherewithal for it as one does in expending it gainfully. Don't become a fieldworker if you don't want to beg for funding, unless you're independently wealthy.