There's this guy Brandon who has a project called Humans of New York. He posts a couple of street portraits every day on Facebook together with short interviews. His project has now turned into a bestseller photobook. Take a look, his method is to approach people softly and respectfully as far as I can judge from the short films he has posted about his method.
Another person who has had great success as a modern street photographer is The Sartorialist. He is into fashion and makes portraits on the streets of various cities around the world. I guess he works fulltime on his blog, gets invitations etc.
So it is not impossible when you can communicate a strong reason why you want the portrait, I guess.
I have tried street portraits myself when I lived in Moscow, and I most often tried not to be noticed. I am not sure I feel totally right about it though. Some people seemed annoyed.
erikpetersson.livejournal.com (go backwards in my blog one or two pages)