I guess this is more of a question for people who live in more western countries because those are generally more ethnically pluralistic in their cultural makeup but, do you generally look to meet people who are of the same culture as you or different, or does it not matter at all?
My girlfriend was born into a different race on a different continent - doesn't matter to us.
I don't think it should ever be a black-and-white... 'no, I will not, for she is of that race'. Certainly, sometimes you won't get on with someone for reasons that largely come down to cultural differences. And that's fair enough. But don't rule anyone out automatically without giving them a chance.
In theory, no. In practice, yes. Why do I say that? I think it's easier to understand and relate to someone who comes from a similar background as you. Culture just happens to be one of many factors that can come into play.
I am not sure if the U.S. was included in your question about western countries, but since that is where I have lived most of my life that is what I will use as an example. As far as culture goes if you were born here (any state) then your culture really is a combination of a few hundred years of different people living on the same body of land.
There are certain things groups of people try to hold onto such as ethnic food, religion, and language. But ultimately most foreign cultures are secondary or (watered down), with Americanism with each generation.
And as a way of putting things into perspective, most people eat fast food, watch tv, celebrate xmas and thanksgiving, and are reluctant to state problems with our government.
To actually answer your question, I look more at the individual more so than their family background or history. For me I am an equal opportunity guy. Not even language would be an issue for me, if I was interested enough in a person I would enjoy learning ways to communicate with them.