Behold, the motherlode of free old-time-radio goodness. Start [and maybe end] here.
Botar is chockablock with OTR MP3 goodness; finding the fairly rare Frankenstein serial adaptation is a pleasant surprise.
I'm still branching through this Shadow fansite's radio links page.
Free-otr.com has a frustrating interface [for us impatient Firefox/DownThemAll users, at least] but a rich archive, especially the Andy Griffith and Jean Shepherd shows and "Audition Show" [pilot] for THE VOYAGE OF THE SCARLET QUEEN.
Then there's always Archive.org's swell OTR archive, which offers the big plus of being able to download an entire archive as one massive zip file.
Tokyo Rose, the radio Siren who never existed, RIP. [More WWII A/V from that site can be sampled here.
Radio, but not that old: I recently rediscovered SCHICKELE MIX, a great but cancelled syndicated NPRI series by "the PDQ Bach guy" about his marvelous, inspiringly open and broad view of music. [You can find out when/where MIX airs here.]
Not radio, but old-time; one massive wax-cylinder-record archive. The "Jewish Ethnic Humor" cylinders aren't as funny as I had hoped, but the Hawaiian music is purty.
Not radio or as old: A remarkable bit of scholarship on the "Rock Around the Clock" guitar solo, played by session musician Danny Cedrone.
Not radio or really old-time, but more MP3 sites: Scarstuff, which is packed with great old Halloween and other pop-culture record rips; Gutenberg.org's human-read audiobooks; Play It Again Max, boasting a lot of groovy OP vinyl.
Not radio, but old-time: Holy crap, a whole lot of banned theatrical cartoons!