Hi, this is a very interesting topic.
To answer your question - I would say the first thing you need to do is to decide what element of culture you are trying to teach? Are you trying to teach about an actual event (e.g., American Pie (1971) by Don McLean about a plane crash that killed Buddy Holly and several other musicians), a movement (e.g., Bob Dylan = peace and anti-war movement), or even music as it has helped shape culture (e.g., Elvis, Michael Jackson, or Jazz).
Once you have decided the angle you wish to take it could be wise to select songs that are both age and level (English level and school level) appropriate. There is no use using a song where the students don’t actually understand the importance of the message even if they understand the English (e.g., is there any value in teaching, say, young Korean EFL students about Live Aid/ Live 8 when they have no concept of economics, poverty, or protest movements?)
Right, now we have made appropriate choices it is time to engage the students. Get them to relate the song to their beliefs or culture etc. Get them to play with the new words and metaphors etc. Get them to engage with the song by singing, dancing, or acting out the story.
Culture is alive – so teach it like it is alive.
I hope this helps,
Josh