Many attempts have been made to ground morality outside of a personal God, but all fall miserably short. At best, non-theistic ethical systems offer a rationale, or principle by which one can justify a system of prescriptions and proscriptions, but in what do they ground the rationale? The guiding principle may provide for a consistent system of ethical thought, but just because a system is consistent does not mean it is true, or that anyone is obliged to adopt it. Offering a rationale for saying one ought to do X is very different from grounding that moral imperative itself.
The only way to ground a moral imperative is to anchor it in some transcendent source. Any system that is grounded on principles created by man cannot transcend man because it has no objective value. It is entirely subjective; a social convention; morality by the people, of the people, and for the people. Society could choose to adopt a totally different rationale that supports a totally different set of prescriptions and proscriptions without violating any moral truths, because non-theistic moral systems are not representations of moral reality. Indeed, there is no such thing as moral reality (moral anti-realism). In the end, non-theistic moral systems provide no ontological basis on which to hang objective moral rules, and thus offer no compelling reason to abide by the rules of the system.
Some atheists believe objective moral rules exist as part of the fabric of the universe (they are moral realists). These moral laws are said to exist as inexplicably as the laws of nature itself. If so, the grounding problem would be solved, because there would be an objective basis for moral prescriptions and proscriptions. But why think we are obliged to align our lives with these moral rules? Obligations are grounded in relationships, and relationships entail personal agents. If moral rules are not grounded in a transcendent moral being, it makes no sense to think we are obligated to follow them. They can be safely ignored without enduring consequence.
But what if we chose to follow them anyway? Would it matter? No. Our moral choices would be insignificant because the finality of the grave allows for no moral accountability. I will not be rewarded for having obeyed the natural moral laws, and you will not be punished for having ignored them. The outcome is the same. In the end, it becomes meaningless. A moral realism that is meaningless is no better than a moral anti-realism that is meaningless. Only theism can ground objective moral values, our duty to submit to those values, and supply us with a rational reason to fulfill our moral obligations.
I should make it clear that the question is not whether non-theists can recognize objective moral laws apart from belief in God, or even keep them apart from belief in God. They can, and do. The question is how they can make sense of that which they recognize, and make sense of that which they do. Apart from theism, I think the answer is negative.