For U.S. books published between 1923 and 1963, the rights holder needed to submit a form to the U.S. Copyright Office renewing the copyright 28 years after publication. In most cases, books that were never renewed are now in the public domain. Estimates of how many books were renewed vary, but everyone agrees that most books weren't renewed. If true, that means that the majority of U.S. books published between 1923 and 1963 are freely usable.Link (Thanks, Frances!)
How do you find out whether a book was renewed? You have to check the U.S. Copyright Office records. Records from 1978 onward are online (see http://www.copyright.gov/records) but not downloadable in bulk. The Copyright Office hasn't digitized their earlier records, but Carnegie Mellon scanned them as part of their Universal Library Project, and the tireless folks at Project Gutenberg and the Distributed Proofreaders painstakingly typed in every word.
Thanks to the efforts of Google software engineer Jarkko Hietaniemi, we've gathered the records from both sources, massaged them a bit for easier parsing, and combined them into a single XML file available for download here.