I am a red-blooded American white boy. In fact, my black friends tell me that I am about as "white" as anybody could ever hope to be. Whatever that means.
But I will offer my opinion. I think being of a different race or having a different accent is not really that big of a deal. The important thing is that your patient be able to identify with you. That doesn't mean having the perfect Midwestern accent so everybody thinks you were born in Ohio, but it does mean speaking English very comfortably so people don't have to make a special effort to understand you.
For your patients to identify with you, you must also become American. This is harder to explain. It does NOT mean giving up all traces of your religious and cultural heritage; you just need to make an effort to assimilate.
I don't know how to explain this well at all. I have a good friend who has possibly the worst Vietnamese accent I have ever heard, but he is so American that I have a difficult time imagining him living anywhere else but in the good ole USA. He doesn't make other people try to pronounce his Vietnamese name, but goes by a close sounding American equivalent.
I have met other Vietnamese friends of his that speak much better English than he does, but they always seem out of place and a little awkward in American society. I like these people, but would not feel comfortable with these people as my doctor. There is something too foreign there that I just don't understand. Maybe somebody else can explain this one better.
I hope I don't come across sounding racist; just trying to offer an open and honest opinion.