Logic. Moral philosophers, even those in the employ of the Catholic Church, rarely base their arguments on appeal to authority, because you can't convince non-believers to adhere to your code of ethics if the only backing for it is "God said so".
In a future time dominated by atheism, certain actions may still be morally required (or morally forbidden) because of similar logical argument. These logical constructions generally fall into two camps; deontology (an act is inherently good or evil based on its adherence to or divergence from a rule system) and moral relativism (an act is only good or evil relative to the context in which the action occurs, such as the consequences of the action or the scope of the observer). A third, moral nihilism (no act is good or evil for any reason) is not useful to us here.
Let's consider polygamy. From a deontological standpoint, specifically Kantism, polygamy is wrong because it creates a logical contradiction if the practice were made universal. My ethics professor introduced the Kantian model to us using a word I will not repeat as such, but the process is to "Formulate", "Universalize", then "Check" for "Kantradictions" (contradictions).
So, let's formulate the rule: "It is morally acceptable for a man to marry as many wives as he wishes as long as one woman is married to one man" (the traditional definition of polygamy; polyamory is more flexible with regard to the number of men and women in the relationship). Then, we universalize it; every man marries as many women as he wants. Now, we check for contradictions. Well, the human species is roughly 50-50 with regard to gender (it varies regionally due to a number of factors, and globally the average is just slightly male-heavy at 101 men for every 100 women), so if every man got as many women as he wanted, and a woman could only be joined to one man, some men wouldn't get any women. Those men may want multiple women, and should, by the rule, be allowed to, but for practical reasons, they can't because there aren't enough to go around. This maxim making polygamy morally acceptable, then, creates a logical contradiction - if some men get more than one woman, other men who want one or more wives can't have any - and polygamy therefore must be morally unacceptable, instead favoring the more equitable "one woman per man" monogamy rule.
From a relativistic standpoint, such as utilitarianism, polygamy may be acceptable, on a case by case basis. The taking of more than one woman as a wife by a man should not cause more total harm than it does good. There's an "ethical calculus" involved here; we must weigh, for each case of a man wishing to marry an additional wife beyond his first, how it will affect the man's existing wife or wives, how it will affect any current or future suitors, and how it will affect the platonic relationships between this family unit and the rest of society. This calculus is difficult in the general case, but based on knowns, we can predict that we would still run into the "too few women" problem; If a woman is taken as a second or third wife by a man, it reduces the chances of another man in the community finding any wife at all. This would be a considerable long-term negative for the practice that may lead to a similar conclusion from the relativists as the deontologists.
Most other "sexual deviances" that are immoral on religious grounds could be argued against on more practical or purely logical grounds as well. Homosexuality could be discouraged as increasing the potential for disease and as being unable to produce an offspring genetically related to both of the partners. Extramarital sex can be determined to be immoral because it also increases the spread of disease, and produces children whose parents are unwilling or incapable of caring for it, thus increasing the burden on society as a whole. Incest produces undesirable genetic mutation. Sexual abuse of children causes a host of mental and developmental issues.
Virtually all of these logical arguments have a counterargument; in your world, the simplest hand-wave is that society as a whole finds any counterargument unconvincing. This could be due to some revelation between the present day and the future that convinces society as a whole that cis-heteromonogamy is the only way to go. For instance, the AIDS epidemic set gay rights back at least 20 years, and effectively muted the general sexual liberation that came out of the Summer of Love. If a disease even worse than AIDS, maybe some sexually-transmitted ebola-like virus with AIDS' vaccine resistance, were to ravage our near-future society, you might find public support for anything more sexually liberal than serial monogamy vaporizes in short order, as promiscuous people are many times more likely to catch the disease.
