What background is necessary to do research in computational linguistics? I'm getting ready to enter graduate school, and I would like to do PhD-level research in computational linguistics and natural language processing. I know that these two fields are very interdisciplinary and draw on various subfields of linguistics, computer science, math, and statistics. I also know that research in comp ling and NLP is done in different departments at different schools, with some schools having it in the CS department, and some in the linguistics department.

Ideally, what background knowledge should someone have in order to do research in computational linguistics or natural language processing? Which areas of linguistics, computer science, math, and statistics are necessary or helpful in studying comp ling and NLP, and is there one field among those four which is overall more necessary than the others?

(Note: per this question from Linguistics.SE, the distinction between comp ling and NLP is pretty blurry, which is why I mention both in my question.)
Linguistics Ph.D. admissions is unusual in that no particular background is necessarily required.  The reason is that many institutions do not offer a linguistics major, so the pool of linguistics majors is small.  In addition, linguistics is highly interdisciplinary.

To answer your question more specifically, if you want to enter computational linguistics, some programming ability is essential.  A bachelor's degree in any of the disciplines you mention may be sufficient.  I would also add psychology, anthropology, and foreign language backgrounds as potential entry points for a linguistics Ph.D.  You must also be prepared to learn statistics.