It worked for King, it worked for Ghandi. Do you really think there could have been something like the civil rights laws -- born primarily of white people -- if the primary black person had been X and not King?
It is patently absurd to give MLK's non-violent movement 100% of the credit for civil rights reform. The political situation during that time was unbelievably complex, and the changing attitudes of a turbulent nation cannot be so simply explained.
Yet it was Ghandi himself who, when asked if his peaceful tactics would have worked against Hitler, said, "probably not."
I'm not suggesting that white people are literally Hitler, but simply that there's a time and place for everything, including peace and violence. And I say that as someone who has never called for violence on anything ever.