Is it true that there is no absolute morality?

[Neo-Scholastic]

That breaks down the moment you encounter a culture with different values that you would consider 'deal-breakers'.

Exactly right, but also where our culture might be heading. Purely secular moral systems (mostly based on Marxist theories) do not have the most impressive record during the 20th Century. Or earlier for that matter. The French Revolution slogan of Liberty and Equality was quickly put into practice with the guillotine. Today, in the West, "bio-ethicists" like Peter Singer and others have started arguing for post-birth abortions, terminating the mentally disabled, and flirting with eugenics based on 'rational' principles. Personally, I don't see the high rates of abortion and the caviler attitudes about it as a sign of progress. It remains to be seen if these secular values will persist long after becoming disconnected from the Christian cultures in which they arose. The values of Western European and Anglo-American democracies that seem so obvious to humanists could be a historical bubble.