I think that the issue of the subjectivity of morals gets confused with the subjectivity of language. Different people use the term "morality" and its cognates to mean different things.
To some it means the beliefs of various cultures at various times concerning what ought and ought not to be done. Under this definition, it is absurd to deny that morality is subjective, for certainly what the people of a given culture believe is dependent entirely upon what the people of a particular culture believe.
The same can be said for the times when "morality" is taken to mean the beliefs of an individual at a given time.
To some, "morality" means the commands of a deity. If that deity exists, then these values are objective. If that deity does not exist, then, as an objective fact of the matter, nothing is moral or immoral.
To some, "morality" means some objective, intrinsic property built into the very fabric of certain types of actions and character traits. I think this view dominates our culture - not in theory (when speaking theoretically, most would state that morality is subjective), but in practice. This is demonstrated in the way people debate moral issues - each person calling upon evidence and counter-evidence precisely the way they would debate any issue where they assume there is an objective fact of the matter "out there" to be discovered. On this account of morality, I would say that right and wrong do not exist. It is a fiction, a myth.
There are other conceptions of "morality" that do point out objective facts. The act that brings the greatest happiness to the greatest number is a fact - though, unfortunately, not knowable and, thus, not a very practical ethical standard.
So, morality is subjective and objective, relative and absolute. It is all these things because the term "morality" means different things to different people. And language IS subjective. There is no natural law that requires that a word mean a particular thing. Language is invented, and nothing prevents different people from giving different meanings to the same word.
Subjective morality might be an interesting area of study for the sociologist or psychologist, but, for all practical purposes, they are useless. The provide us with no way to solve interpersonal or intercultural disputes. For this, we must look for a morality (for morally relevant facts) outside of individual or intercultural differences.
Religious and "absolute" moralities are built on myth, and have no place in real-world decision making effecting the lives of real people. (Ayn Rand's system qualifies under this category, for it is an "absolute" morality built on the false assumption that values are woven into the very fabric of certain states of affairs).
Utilitarian moralities are objective and interpersonal/intercultural. Though simple utilitarianism is impractical, there are different versions. (Note: just to confuse matters, there are some utilitarian moralities built on the assumption that certain psychological states - pleasure, pain - have an intrinsic "absolute" value to be maximized. These utilitarianism's are as mythical as any other religious/absolutist morality.)