A: They take them, but 99% of the time they don't actually play them.
Q: Why not?
A: Most playlists are determined well in advance.
Q: What do you mean?
A: Radio stations "test" songs using auditorium testing (or other means) and develop a playlist of songs that score as "familiar" and "likable". From there, a station has a list of (usually 300) songs they deem "safe" that gets played.
Q: So why do DJs always say that they'll "try to get that song on"?
A: Because they're being nice.
Q: So there's no such thing as a request that actually gets played.
A: Not really although people tend to request songs that are current and would likely be played anyway.
Q: I see.