Hello Sumi,
Well, if you take a look at this listen to this:

http://www.jamaicans.com/childsguide/music.htm

You can listen to the music here:

http://www.jamaicans.com/music/audio/World_Is_A_Stage_56k.wma

When they say the following:
"With me Baby"

The correct English would be "With my baby"
But as it's how people would speak in Jamaica.

Now back to the original question:

If you look at the way when people say
"No man!" Or "No way man!" The adding of man
at the end of sentences originates from Jamaica - I think - I might be wrong or I might be correct. OK man?

;-)
A pun there! Couldn't resist!
So to a woman - "No woman, no cry" - This can be translated to be the following:

"No lady, please don't cry"
I think it's a fair translation...Would anyone else disagree or agree?

Sumi, I'm not Japanese nor am I Jamaican. I have Jamaican friends, and I was started to listen to Jamaican music when I was young.

See you around!
ROuNIN