After slowly working my way through all the e-mails sent in response to my plea for information on copyright and public domain law as it pertains to audio recordings, it's clear that the issue is a good deal more complicated and murky than I had assumed it to be. Although I still feel more confused than enlightened, here is, I hope, a reasonably lucid summary of what I've learned.
Simple stuff first: Anything recorded in the U.S. prior to 1923 is in the public domain. After that, it becomes ridiculously convoluted. Any particular performance of a piece of music is covered by two copyrights: one for the composition, another for the specific recording of the composition. However, it was not possible to copyright a sound recording prior to Feb. 15, 1972. As audio archivist David Seubert explained to me, "That doesn't mean that all recordings before the 1970s are in the public domain. They are covered by state and common-law copyright, but even among experts there is disagreement on what this means. The conservative argument is that all sound recordings are copyrighted until 2067, when the pre-1970s recordings are set to enter the public domain according to the Sonny Bono Copyright Extension Act of 1998. However, a more liberal interpretation is that since they are not covered by federal copyright, they are essentially fair game, which is the reason you see many older recordings at archive.org and on reissue CDs."
Indeed, to judge from both the e-mails I received and the independent research I did on the Internet, there isn't any kind of consensus on how to interpret the tangle of copyright laws that apply to sound recordings. Assuming that the truth lies somewhere between the two extreme interpretations (that nothing post-1923 is fair game, or that everything pre-1972 is fair game), there is, unfortunately, no simple way to determine whether a specific recording is still under copyright, since there isn't any database that catalogs copyrights.
Grateful thanks to the many people who took the time to write in with explanations of how these laws work.