A large part of the difficulty with this problem is that it is not well-posed. The relation "$A$ is married to $B$" is ill-defined, and depends on a great deal of non-mathematical interpretation and assumptions. For example, many societies are polygynous (one man, several women). In such a society, if a man is married to several women, are those women considered to be married to each other? The nature of the relation defined by the phrase "$A$ is married to $B$" depends on how this question is answered.
It is also possible to conceive of a society where this relation is not even symmetric—perhaps several women can be said to have married one man, but that man cannot be said to be married to all of those women. I have in mind the vows that Catholic nuns take, in which they are said to marry Jesus. Is that version of marriage symmetric? I honestly don't know the answer to this question, but it is the kind of thing which pops up.
So, in answering the question "Is marriage a transitive relation?" it is first necessary to define what "marriage" even means. For the purposes of this question, I am going to adopt the following, very narrow definition, which more-or-less conforms to current legal definition of marriage in the US:
In the United States, marriage is a legal arrangement between two individuals. When two individuals enter into a marriage, they marry each other (hence if $A$ is married to $B$, then $B$ is also married to $A$). Moreover, if $A$ is married to $B$, and $A$ is married to $C$, then $B = C$ (this is the legal requirement that a person may not be married to more than one other person). Finally, a person cannot be said to be married to themselves (the relation is irreflexive).
Under this definition, the marriage relation is not transitive. If $A$ is married to $B$, then symmetry implies that $B$ is married to $A$. If marriage were transitive, then $A$ would be married to $A$. However, we have defined marriage to be irreflexive, hence $A$ cannot be married to $A$, which implies that marriage is not transitive.
That being said, let me reiterate that this is not really a question about mathematics. It can be made into a question about mathematics by writing a reasonable definition of marriage, but the answer is not unique. For example, maybe marriage is simply illegal or doesn't exist at all (the Na of China are a purported example of the latter). If marriage is doesn't exist, then marriage is transitive by vacuity. Or maybe society is polynadrous, but marriage is intransitive because the several brother-husbands of a given woman are not considered to be married to each other. Indeed, given any possible permutation of the axioms of an equivalence relation, I can probably describe a hypothetical (or real) society in which there is a version of marriage satisfying those axioms.