Most people haven't read the original untranslated versions in order to understand them better, but a lot of academics, such as classics professors, have. I've learned about Greek culture from a few professors who would discuss at length how the Greek conceptions of, say, honor or cunning differed from our modern conceptions. But if they were also of the impression that the ancient and classical Greeks did not have a concept of morality, then that would have been a very conspicuous and relevant omission from their instruction. So I'm inclined to suspect that this is a minority interpretation.
I was stunned to read the accounts quoted below. They're claiming that the notion of morality - in the sense of there being a special category of things that you should or should not do for the sake of the things themselves being inherently right or wrong - might not only be a recent invention, but also an incoherent one. Even when I had read debates about e.g. moral realism, I had always understood even the moral irrealists as acknowledging that there are genuine moral attitudes that are fundamentally ingrained in people. But I hadn't ran into a position claiming that it was actually possible for whole cultures to simply not have a concept of morality in the first place.
I'm amazed that I haven't heard these claims discussed more. If they're accurate, then they seem to me to provide a strong argument for both deontology and consequentialism - at least as they're usually understood here - to be not even wrong. Just rationalizations of concepts that got their origin from Judeo-Christian laws and which people held onto because they didn't know of any other way of thinking.