Cultural norms often go far beyond morality. Morality has nothing to say, for instance, about whether men should grow beards or women should wear veils or makeup. So what is accepted or disapproved of in a culture often is not a matter of moral significance.
Next, cultures vary in how they express things that do matter morally. Morality may give us a reason for expressing our regard when we meet someone, but it doesn’t specify whether you should do this with a handshake or a bow. Morality may give us a reason for following the rules of the road that keep us safe, but it doesn’t specify whether you should drive on the right or left side. Lots of cultural norms are matters of convention. But these are often just different ways of achieving the same morally good results.
And next, it is certainly possible for cultural norms to get morality wrong. I can’t make axe murdering morally fine just by saying so. Neither can a whole bunch of people that constitute a culture. Of course we can’t just judge a culture by the standards of our own culture and thereby hope to get morality right. Where there is a moral difference, the problem might be with our own culture. But our grounds for objecting to a culturally endorsed practice might be more principled than just “that’s not the way we do things around here.”
Many cultures, for instance, are pretty hard on people who are gay. This included ours not so very long ago (and still does today far too often). But we have moved in the direction of being more respectful of gay people for fundamentally moral reasons (our more homo-phobic standards of the past certainly didn’t move us in this direction). More and more people have come to see that we lack any moral justification for discriminating against gay people. And the moral problem with doing so is not so hard to see by exercising our moral imaginations (imagine what it would be like for your love to be forbidden).
And DCT
Ah, good. The first thing I want to point out is that it doesn’t sound like you are a Divine command theorist. You are taking morality to be grounded in God’s nature, God’s goodness. And you are understanding his commands and communicating truths that hold not just because God says so, but because God is good. So you are taking morality to be grounded in something other than mere say so and offering a kind of theological moral realism.
Now, do we really need God to tell us murder is wrong in order to get the point. I don’t think so. And I don’t think Christianity requires this idea either. Christianity has that we were created in the image of the Divine. This suggest that having a perhaps limited and imperfect moral conscience is part of our nature (I doubt being created in the image of the Divine is meant to suggest that God needs to clip his toe nails from time to time).
Further, the idea that we have a God given moral conscience of our own is much more in line with Christian ethics than DCT. If our motivation for doing the right thing is just that God says so (perhaps backed up with the threat of hellfire and damnation), the morality is mere prudence or self interest. But Jesus teaches love for our fellow man. The only way this makes sense as a moral teaching is if our moral motivation is internal, where we have our own reasons for being good (our own loving regard for the value of our fellow humans). The morality Jesus recommends isn’t just a matter of following rules or obeying orders.