Historical Voices (Michigan State University, Matrix), including the sub-galleries:
American Voices (including the Flint Sit-Down Strike Audio Gallery)
Earliest Voices
(In addition, the large holdings of the Vincent Voice Library at MSU are catalogued and searchable on-line, though not all are available for download and listening.)
The Library of Congress has several sites devoted to preserving American history via sound; see especially:
Voices from the Dust Bowl
Edison Sound Recordings
Voices from the Days of Slavery
American Leaders Speak
Working in Paterson
Free Information Society (mp3 recordings of speakers ranging from Joseph Goebbels to Mahatma Gandhi)
Famous Speeches in History (The History Channel)
American Rhetoric (speeches)
SignalAlpha: Historical Audio and Video (has sections on old-time radio programs and advertising, World War II)
Documenting the American South, Southern Oral History Project
U.S. Labor and Industrial History Audio Archives (University of Albany)
British Library Sounds, Oral History segment
Radio Days (primarily programs, but advertising is also heard)
For historical music recordings, see the Library of Congress's National Jukebox.
In addition to historical recordings, several sites offer audio of discussions of historical topics. See, for example, "Talking History" and "Backstory with the American History Guys."