Until two years ago I had always been an atheist, but with basically the same morals I've got now. And you know why? Because doing good has always made me feel good. And doing bad things has always made me feel really bad. Even if I wouldn't get rewarded once I die. And most people are this way, including atheists.
There are, of course, always bad people that will try to do whatever is bad, but this isn't influenced by religion. When good people read the Bible, they will mainly notice and preach the teachings of Jesus about love and doing well to your neighbors, while bad people will mainly notice genocide, punishing people etc., and preach that. And the same counts for atheists. While the vast majority of atheists want the world to be peaceful and harmonious, there will always be a part that hates love.