I love the spoken word over some beautiful background music. However, this seems to be the kind of thing that many people dislike or dismiss very easily. Maybe it's because I've always enjoyed poetry, but I think it's a nice change of pace from singing. Of course I could create a huge list of bands that have done this in the past but for now I'll focus on the recent ones.
EMA - The Grey Ship A little more sing-songy than spoken, this is still a really dark and impressive track. However her video below for "California" is more spoken and just as much an open wound. Check out her website for more and expect a full length later this year. She's bound to be much talked and buzzed about. She has that "different je ne sais quoi" quality working for her.
Savage Nomads - Pineapple reminds me of The Streets, with a pretty chorus. Their debut album is expected to be released in June.
The Savage Nomads - Pineapple (unmastered) by Alaska Sounds
Error Operator - Follow This cinematic song borrows lines from the movie "Network" dating back to 1976. Error Operator puts out an endless stream of electronic music and I highly suggest following on soundcloud. I've played this song unbelievably loud all throughout August and September, I suggest you do the same. "All I know is first you've got to get mad. You've got to say 'I'm a human being god damn it! My life has value!." If you yell this part of the song your overall disposition will improve by 60%. Seriously. Music is better than therapy, that's my motto.
This is a rather brief list...I ran out of time. Leave your suggestions in the comments section.
Starf*cker's song "Quality Time" song from Reptilians has a decent spoken intro sounding like it's from a science video.