What background is necessary to do research in computational linguistics?
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.