Dr. Harris, as a member of the new atheists, is deeply antagonistic toward organized religion. His organization, Project Reason is established to spread secular values and oppose organized religion. His book on morality is an attempt to both provide a secular, scientific, and rational basis for moral values and to refute the view that religion is necessary to provide morals and give meaning to human life.
He actually creates something of a strawman argument when he speaks of the issue. He states that Christians teach that a person can't be moral without belief in God. Actually the Bible says the opposite. Paul tells us that Gentiles (those with a wrong concept of God or no concept of God) instinctively follow the moral principles of God's law, because it is "written on their hearts." (Rom. 3:16) This notion, that morality is built-in to human nature, lies at the foundation of the principle of natural law and Christian ethics. For this reason, an atheist not only can be moral, he/she actually wants to be moral. It is one of the ironies that escapes Dr. Harris, his argument against God and Christianity is grounded in his moral sensibility (Religion is the source of war, evil, and oppression in the world), yet he has no explanation for how an entirely amoral, natural process such as evolution could have produced these moral inclinations in man. It's not that they don't try, which explains much of what the book is really about.
The moral system of the book is built upon classical utilitarianism, which holds that morals is about creating or protecting well being for the largest number and likewise, avoiding pain and suffering for as many as possible. Utilitarianism has two serious problems that has been demonstrated in its use in human history. The first is that it's logic tends to de-value the individual in the debate over good and evil. The well being of an individual (or a few individuals) is not as important as the well being of the larger community. This results in two problems, first, if you are not one of the lucky ones who is part of the "larger community" you get to be the sufferer. In other words, it leads to justifying the elimination or exclusion of those not fortunate enough to be the greater number. Second, it endorses a moral reductionism. Immorality is bad for very pragmatic reasons, it harms society or makes it difficult for people to trust one another, rather than because it is just wrong. So, in the extreme, drugs might be considered wrong because of the harm they do to communities but not because of the harm they inflict on the individuals who take them.
The danger in any attempt to create a rational foundation for morality is that human beings are much too good at rationalization. We are wonderful excuse makers and at using our minds for all types of self-justification. Look at the response of most politicians when they are exposed and accused of wrong doing. It is always some form of "I'm innocent of all charges." The larger problem with utilitarianism is that it is too easily set aside and rationalized or worse, twisted into the justification of actions that are both evil and deeply harmful. As a case in point, I would simply direct you to eugenics and the forced sterilization of thousands of people in the 1930's in the name of science and utilitarian moral principles.
The danger in invoking science and reason in morality is that these have no transcendent foundation. They are based on limited human knowledge and experience. We are too short sighted to understand the good or the bad of chosen behaviors or lifestyles. Only decades later, after we have weakened the institution of marriage, do we discover the damage the arises from no-fault divorce and single parent households.
Worse, since the enlightenment, we have rejected much of what human society assumed to know from history and thousands of years of human experience. We think we can change the rules about marriage and the family simply because we are so advanced, but in doing so, we are not acting reasonably, we are rationalizing. I believe that later generations will look back at our time as a time of outrageous arrogance. As we have sought to create a "new" morality that is really the old immorality, and that will, in the end, not acheive the greatest good for the greatest number, but will produce massive damage to the majority that always ends up following the cultural pied pipers.
Our own society was not founded on utilitarianism, even though it was a powerful concept that emerged from the enlightenment. We value the individual and we seek to protect minority rights. As we state in the Declaration of Independence, "All men are created equal, endowed by their Creator with certain inalienable rights..." Many have argued that we were not (or were) established as a "Christian" nation, and I would be one who would say that we were not in the direct and intentional sense. The founders understood that we were religiously diverse, even at the end of the eighteenth century. They wanted a free and just society open to all, but founded upon Christian principles. There is no question that in regard to the value of individual and the importance of a transcendent moral structure for society, that they were influenced by Christianity and not by the enlightenment thinking of their day. For not only do they invoke the importance of the individual, they invoke "inalienable rights" these rights are not given by the state or by society at large, they are inherent rights based upon an inate structure of right and wrong built into the universe itself.
Based on the success of the great American experiment, one could assume that a transcendent view of morality gives better results and fits human nature far better than the innovations of utilitarianism. Like so many attempts to create a secular and rational morality this latest attempt by Dr. Harris crashes into the rocks of human fallibility. We are Romans 7 creatures, "For I have the desire to do what is good, but I cannot carry it out." (Rom. 7:18) Not only do we need a set of transcendent moral values, we need the grace of God to keep them.