If you think the morality of ancient cultures (Ancient Rome is a good example) had anything to do with their downfall you're probably wrong. You could maybe argue that their morality led to conflicts, for instance slave rebellions and such, but beyond that you would have a hard time making a decent argument that morality had anything to do with it. You could maybe argue that greed had a lot to do with it, and you'd probably be right, but that's really something that is inherent to humans/survival of the individual. And even then, it isn't necessarily the greed of the cultures that fell but rather the greed of the cultures that killed them and took their stuff.

The best argument I can make against morality being a factor, though, is to suggest a hypothetical situation. Imagine there were an ancient culture that was perfectly moral, never stooping to cheap tactics in war, the use of slave labor, etc. They would probably be one of the first to fall, considering the less moral cultures of the time would have more tools at their disposal and more power. I'm pretty sure most or all of the really big empires were built on the backs of slaves, including our own.