The National Black Republican Association thinks so:

So is it true?  I guess it depends on what your definition of a Republican is.

If you could say that MLK was a Republican, and I think it’s a stretch, he’d be a Mike Huckabee-style Republican.  Very socially conservative, but very much a populist as well.  It can be hard to guess about King’s feelings on search fiscal policies, but I doubt he had much in common with Milt Friedman.  Plus, despite King’s famous speech where he said that people should be judged by the content of their character instead of the color of their skin he was very much a supporter of affirmative action policies.

So would King be a Republican today?  I doubt it.  King was right about much of the civil rights problem this country had when he was alive, and he was undoubtedly very socially conservative which doesn’t contrast well with the social liberalism embraced by today’s Democrats, but ultimately I doubt he would buy into the GOP’s core fiscal values.