There are two kinds of cultural relativism, and I think the distinction between them may explain why the same confusion comes up whenever people discuss the moral argument in debates and discussions.
The two kinds are descriptive and prescriptive. Descriptive cultural relativism is just the observation that different cultures have and do, in fact, hold to different standards of moral behavior and values. Sometimes they might agree in what personality traits they think are virtuous, but they differ in the degree of value they place in each of the virtues.
Prescriptive cultural relativism is the view that whatever moral norms a society accepts constitute our moral obligations. In other words, people are obligated to live and behave in conformity what their own culture's moral standards.
The descriptive view isn't that controversial. The only controvery is the depth to which different cultures disagree. I think there is far more agreement than disagreement, and that most of the disagreement is superficial. It can be accounted for by looking at the underlying moral premises and how different cultures solve moral dilemmas. But hardly anybody denies that there are at least some differences in the values and morals that different cultures hold to.
Prescriptive cultural relativism can be reduced to moral objectivism because it depends on the supposed universal principle that each person ought to behave consistently with the morals of their society. Since that principle is universal and transcendent, it's not culturally relative even if everything else is. If there is no universal moral presciption about living consistently with one's own cultural morals, then prescriptive cultural relativism would reduce to individual subjectivism.
One of the primary objections people often bring up against objective morality is to point out that different cultures (and the same culture over different times) subscribe to a different moral point of view. I used to attribute this kind of response to a confusion between moral epitsmology and moral ontology. I would grant that, yes, different people have different moral beliefs, but the moral argument isn't concerned so much with the beliefs as with the ontological reality of moral prescriptions. So the standard response to this objection was to say, "Well, people have also had disagreements about the shape of the earth, but doesn't mean there isn't an objectively true answer to the question of what the shape of the earth is."
But I wonder if, rather than being a confusion of ontologogy vs. epistemology, if the confusion is, instead, between prescriptive cultural relativism and descriptive cultural relativism. Descriptive cultural relativism tells us nothing at all about the ontological existence of objective morality. Moral objectivism (or moral realism) is perfectly consistent with descriptive cultural relativism. It's only prescriptive cultural relativism that poses a challenge to moral objectivism. So I think this distinction ought to be fleshed out when it looks like there's a misunderstanding going on in these discussions. I suspect that when people raise the observation about cultures disagreeing on morality as a challenge to moral objecivism that they are confusing the two kinds of cultural relativism. They are conflating one with the other by treating descriptive relativism as if it were prescriptive relativism.
I suppose the best way to handle the situation is just to ask questions for clarification on what the other person's objection is. Maybe they misunderstand your position and think you are claiming that moral beliefs are universal or that everybody holds to the same moral standards.