Because a previous attempt to explain morality resulted in an untenable skyhook, here's my bumbling attempt to explain a foundation for morality that doesn't need some magic explanation. The common argument against naturalistic morality is that without a mind moral facts cannot exist, hence any attempt to put forth morality without a moral giver by necessity has to be subjective. While I think there are problems with the notion of a moral-giver (see: The Euthyphro Dilemma), the lack of any naturalistic account is something that is a concern to many.
What morality is
In one sense it should be obvious that morality can be entirely embedded in naturalism, it concerns how we ought to behave. And aside from behaviours that are aimed at the potential next life, behaviour affects here and now. If you steal from someone, while that may result in you reincarnating as a leper in this life the action affects both you and the other person.
Irrespective of whether there's a supernatural, there is still the natural. And since no person is an island, there will inevitably be interactions. And since individuals can have conflicting goals and desires, there will be disputes. Our actions affect others and their actions ours, so it's not hard to see why there's a need for distinction between how one does behave and how one OUGHT to.
I can't think of any mammal that has no social interaction whatsoever; mammary glands pretty much guarantee there's some form of interaction. The reason I bring this up is to highlight that the environment has been shaping us for hundreds of millions of years in how we behave and how we interact with others. The implication being that a lot of how to get on with others is part of who we are already, not that the OUGHT follows from the IS.
One criticism of naturalism is the statement 2+2=4. This is not a material fact, so the implication is that one needs a mind to hold the statement as true. The reason mathematics works, it is stipulated, is that God holds the mathematical truth as universal. Which upon the slightest reflection is absurd. When you combine a banana with another banana, it's not that you have 2 bananas on God's insistence. The relationship would exist independent of a mind. It's just that we have a mind that can recognise that fact.
I bring this up because I believe moral facts exist in such the same way. Looking for morality in an atom is about as valid as looking for mathematics. You can't scientifically show what a 2 is, it's an abstract concept after all, but it would be downright absurd to think that it requires anything more than naturalism.
A knife doesn't cut because of the infinite imagination of the designer, it cuts for the same reason a sharp rock does. A rock doesn't need to be designed or have any mind behind it to cut, again it's am emergent property that a designer than look to use. A bird's beak can crush the shell of a nut, likewise so can a chimpanzee with a rock. The only difference is that the rock technology is a sign of intelligence while the beak is just natural selection at work.
Moral facts could exist in much the same way. As animals we need to eat, to not eat is to bring duress and eventually a painful death. This is a fact to contend with when confronted with a starving man. This fact might be weighed along with a whole host of other moral facts, think of a doctor cutting off an arm to prevent an infection spreading compared with someone chopping off another's arm because he didn't like the man. There's loss of the arm and the pain in both cases, yet it would be hard to say both actions are the same. The doctor's taking into the fact that his action would save the patient's life.
If for example sex didn't have an emotional bond to it, that would surely change how people feel about sex outside of a relationship. That there are issues such as the breakdown of trust and jealousy as it stands wouldn't be taken into consideration if they weren't there.
Because no man is an island, and philosophical zombies make for little more than good solipsism, the question becomes how as someone who interacts, cooperates, and competes with others deals with the fact that others they are dealing with are also in that position. We are dealing with the problem that other people may have different or competing interests to us.
A way to think about this is flocking. Certain birds flock, and there is advantage in doing so. But flocking is an emergent phenomenon, no top-down flock-maker to keep the flock together. If birds were just flying randomly then there would be no flock, so there must be some order to it. Yet all any individual bird is doing is obeying a few rules in local conditions and the form of a flock is emergent from that. If any bird doesn't obey these rules and just flies randomly then they could lose the benefit gained from being part of that flock.
Yet what if a bird could think about how best to flock, one that would be able to see the process unfold and see a way that gives greater efficiency and protection. If the bird tried to do it alone then the bird would be put at a disadvantage and the new technique not adopted. But suppose the bird could tell others and other birds could think it through and then take it on board.
As intelligent agents with somewhat of a reliance on culture, we are able to do that. We can reflect on those hard-wired beliefs and even our cultural ones and see whether they are the best they can be. This can be done not just for our own ends (though we can do that) but beyond that too. By having others follow particular rules it could lead to greater societal outcomes.
Of course even if there were moral facts and we can recognise the best of them, it still leads the question as to why we should follow them? The answer should be pretty obvious, it's in your own self-interest to do so. Two reasons for this: personal and societal. Acting for gain against the moral facts can harm others which can in turn affect your relationships with others, as well as the problem that such behaviour has on society in general. Steal and people will be less trusting, not just of you but of people in general. This relationship isn't only for negative actions but for positive ones too.
The problem of other minds is that other minds have the same dilemma facing them. They have to make choices about what's in their best interest just as much as you do. Any decisions must be inherently subjective, because while they may largely overlap each person is doing what they think is best. The Golden Rule, the bastion of ethical thought exemplifies that the process is inherently subjective isn't necessarily a bad thing. But it does leave the problem of what others OUGHT to do beyond having them act in your best interests.
But questions of motivation are necessarily subjective, the advantage we have as individuals is the ability to influence others and think through our actions, of how we OUGHT to behave. We can modify our behaviour and in part the behaviour of others. Like the flock of birds, we shouldn't need anything more than individuals acting within a small set of rules that the flock pattern is emergent, rather than trying to maintain the flock pattern by getting individuals to to fly within some magical construct of what a flock ought to be.