A friend of mine turned me on to a weekly radio show called Time Warp on 89X, which is an alternative rock station in Detroit (The Only New Rock Alternative, according to their slogan). The show is four hours of early alternative music: new wave, punk, grunge, synthpop, etc. The best thing about it? You don't have to actually be in Detroit to listen to it! You can listen online!
The show is on from 8:00 AM to 12:00 PM on Sundays. It's hosted by the one-named DJ Cristina, who fills the gaps in the music with interesting information about "classic alternative" artists. She also writes a blog called Hang the DJ that includes playlists from past Time Warp broadcasts among the posts about Morrissey, Alison Moyet, The Cure, et al.
And just like when you were listening to the radio when the songs were new, you can make requests to Cristina. I suppose you could call the station if you wanted to be really old school but email, Twitter or Facebook are probably better options.
But don't ask her to play "Stay" by Oingo Boingo. I don't think she likes it.