This is a great question. I would say that morality is weakened when religion is taken away for the most part. I am sure there will be some counter-examples, but generally speaking religion has a moral emphasis. Let me give you a few reason for my rationale.
First, most religions have a moral code (Christianity, Judaism, and Islam). This reason alone will help people to be more moral. If a person grows up in a religious environment, then this code of morality would be deeply ingrained and will leave a lasting impression.
Second, most religions also deal with "eternal matters." In other words, what a person does on earth will have ramifications in the afterlife. Therefore, when the code of ethics, which these religions have is lived out, there is a moral emphasis.
Finally, religion provide a way out of post-modern moral relativism.
Richard Dawkins says in one of his books that the fact that religion may inculcate or strengthen morality nevertheless does not necessarily make it true. I do not know whether religious people are more moral than atheists or agnostics, but I do know that a lot of people who are or claim to be religious have committed countless immoral actions throughout history. Your question is a very good one, and a very important one, but it is one that is difficult to try to answer.