True morality comes from the understanding that actions have consequences, and also from the understanding that others are not essentially different from us, that happiness is not a zero-sum game in which we can become happy at others' expense. Therefore we should perform actions that are likely to lead to good outcomes and avoid actions that are likely to lead to negative outcomes—not just for ourselves, but for all beings involved.
It's possible to construct an objective, pragmatic, consequentialist ethical system that does not rely on any supernatural factors. And that's basically what Buddhism has done. Ideas of karma being inherited across lifetimes is a red herring: it's a way of conceptualizing the ways in which one's actions affect others down the road, which you should care about because you care about all beings, not because it's you in any meaningful sense, as everything you're accustomed to thinking of as you will have passed away.
However, it does ultimately rest on a deconstruction of the concept of the self, which is a hard sell to many people, even though it's hard to argue with the logic. But I'd say that all concepts of morality rely on some degree of flexibility in how one defines the self. Compare Jesus's admonition to treat others as yourself, and that what you do to others you also do to him. I don't think that's just cute figures of speech: altruism as a basic animal trait comes from our ability to blur the line between self and other, which is beneficial to the survival of a community. Fortunately, most if not all people have an element of that in them already; they just need to cultivate it. All immorality, on the other hand, can be traced back to very rigid, limited concepts of the self, along with a lack of concern over what happens to others. We also have that tendency, but it can be reduced with practice.
And I do believe the more expansive view, the one that leads to moral behavior and feelings of solidarity with others, makes individuals happier. It's just that a lot of people don't know that, since they've never really tried it.