Many audio recordings from the early days of radio are now in the public domain. An archive of freely available recordings, sampling many different genres, is available from radiolovers.com.
My initial interest in old radio broadcasts stemmed from an interest in comics, and in particular, the early history of Superman. The serial "Adventures of Superman" ran for 12 years, from 1940 to 1951, lasting more than 2000 episodes and spanning two networks. Many familiar aspects of the Superman mythos (like editor Perry White and cub reporter Jimmy Olsen) were first introduced on the radio, before appearing in print. A discussion of this series can be found at the Superman Homepage.
Radio drama is not dead. The BBC is still producing new dramatic productions for the radio. These and many classic programs can be found at BBC radio 4 and BBC radio 7.