Atheists are capable of performing moral, good deeds, just like a theist. The difference is that they do so without resort to a holy text to tell them what is right and wrong. If you need to resort to a holy text to be told what is moral, good behaviour, then you surely cannot claim that your morality is innate. You are only appeasing your particular deity, rather than doing it for the simple reason that it is the right thing to do.
Religious texts define morality in very blunt and simplistic terms. In Christianity, the Bible states that it is a sin to tell a lie. This is not always a black and white statement. If you were a Jewish sympathiser during the time of the War, and a Nazi knocks on your door asking if you know the whereabouts of any Jews, then according to the Bible, you should reveal the location of the Jews. Surely, in this scenario, it is a lesser moral evil to tell a lie and protect the safety of the Jews.
Atheists consider what is right and wrong on the merits of each individual situation. They have no need to be told what is right and wrong from any text that claims moral authority.
If atheists are capable of behaving in a moral way, then where does this tendency come from. Morality is very closely aligned to empathy. The ability to feel what another person feels. With the exception of psychopaths and sociopaths, most people are capable of feeling empathy towards another person or animal. Empathy is innate. It cannot be taught or learnt from a religious text. If you come across a person who is clearly suffering, such as involvement in a car accident, your immediate response is to help them. This is a response that has nothing to do with what you have been taught from a religious text. It is simply the right thing to do, to help someone who is suffering or in pain.
This leads to a curious question. If an atheist is capable of conducting themselves in a moral way, then is it possible for theists to do the same? The simple answer is a definite yes. Morality and empathy are innate traits, every human being possesses them. Theists, however would much rather accept that these traits come from a higher power, or deity. Atheists on the other hand, would rather believe that we do good things for the simple reason that they are right. Not because they have been told it is right.
Surely it is more satisfying to believe that human beings are born with the innate ability to be good. This also must be true, as where else would atheists get their morality from. This is a better explanation of why morality is so universal. People tend to behave in morally similar ways the world over, irrespective of religious belief, culture or creed. If we accept that morality is broadly similar across such disparate people, then the only explanation is that it must be innate. If that is the case, then we have no need for religion to teach us morality.
I would much rather believe that the human species is capable of great deeds and of being moral, as a trait that we are born with, not something we need to learn and be taught.