They have ever since Kennedy and Johnson addressed race in a big way.
Ethnic groups vary in which party they vote for, but most tend strongly one way or another. And at least one regularly switches parties - Armenians vote for whoever last said that they will recognize the Armenian genocide in WW I. (Conducted by what is now Turkey.) Dole said he would do it in 1996, Kerry says it now. Whether it will happen is in doubt though, Turkey strenuously objects to that recognition and has managed to keep the USA from doing it over the years.
Cheers,
Ben