So I've heard enough arguments in my life as a playwright over who can write what race or gender of characters. Can a white man be allowed to write black people. Can a man write women well or can a woman even be allowed to write a man? It goes on and on with any number of permutations. And after a while I've gotten tired of it. So here's my two cents to add to the argument and we can move on with our lives...
To solve the problem of what race to write your characters try this first. First try to write honest characters and the rest will be easy. That's it. No one has ever gotten mad because someone wrote a complex, gripping character. If when you start out to write your play you're starting with a story that hurts you to the bone and spend the time to write characters that are so real they could be sitting right next to you then no one is going to care at all.
Now you can achieve this honesty in a few ways. First write what you know. A musician that only plays strings doesn't try to compose a tune for a bassoon. Think of Stephen Adly Guirgis' work. His plays have people of all colors and all walks of life. He is white but no one has called his characters into question because these are the people that he knew living in Hells Kitchen New York. The second way to approach this play is to do your research. To follow my music analogy the person doing strings can write a song for bassoon if he reads up on what the instrument can do and how it sounds. Whenever you're trying to write about a culture that you don't know do your research. And always have a reason why you're writing this play. What is so special about this story that you just have to write it?
And lastly please don't go about trying to write a "black" play or a "white" play or etc. Just worry about writing a good and gripping play first and the rest will follow. I think sometimes the theatre has gotten too compartmentalized recently and while a balance is great good stories are kicked to the curb because they don't fit a certain mold. Find a good story and share it with your audience. If you truly engage them they'll follow you everywhere.