One common but bad argument against interracial marriage stems from the fear that it destroys cultures. Mixed Media Watch has a good response to this argument. I would add that racial interaction of any sort, especially intermarriage, should create culture as much as destroying it. Once you stop assuming that culture is the same thing as race, it becomes pretty clear that kids of mixed race can have a culture, and this is so even if both parents avoid continuing cultural traditions of their families. We all have a culture, and every culture is changing. In the U.S., there used to be a black culture and a white culture. Now there's still something to black culture, but there really isn't much of a white culture anymore, just a mainstream culture that includes many historically white elements but has many elements from non-white ethnic groups. If black people were to give up the remaining distinctives (which isn't what I'm recommending; I would recommend giving up only bad elements of any culture), it still wouldn't mean black culture is lost. It would mean some (but only some) of those distinctives would be lost. Many of them would remain on in the continuing culture that contains those and some of the original mainstream features.
But what's really silly about this argument is the idea that mixed race children are being robbed of their own culture if they are not raised in ways that the culture of one side of their ancestry had. Whatever you're going to say about what they're being denied, they're not being denied their culture. It's not their culture unless they once had it. Those who were taken from Africa and made slaves were robbed of their culture. If Sam and I adopt a Korean girl, we're not robbing her of her culture just because her parents would be of two very different ethnic groups that are not her own. That Korean culture was never hers, so no one could claim that she would be robbed of her culture. At best she would be robbed of a culture that might otherwise have been hers, but that's not the same thing. We might say there's some kind of ancestral heritage that she has, but it's not her culture. Her culture is what she's raised with.
Star Trek: Deep Space Nine dealt with this issue nicely in the second season episode "Cardassians". A Cardassian child was raised by Bajorans. I've used that episode with good results in my classes when I've talked about race. A Cardassian child was raised by Bajoran parents. The Cardassians had been the oppressors of the Bajorans, until the Bajorans freed themselves. Some Cardassian children were left behind, and this kid was one of them. The Bajorans took them in. His culture was clearly Bajoran, but his father wanted him raised as a proper Cardassian to appreciate Cardassian things. Sometimes thinking about these things with a no-stakes context like science fiction really helps put things into perspective when you return to the real-life cases.