One word and one phrase need clarification in this question. "Good" and "without God".
I want to look at the phrase "without God". My first clarifying question would be "do you mean 'without God's existence' or 'without believing in God'?" The answer to this question will determine how my unasked question about the meaning of "good" will be answered.
If the atheist answers "without God's existence," then it is quite easy. The answer is "yes" and "no"- both meaning the same thing and being just as valid as the other. Since atheists must base their morals on sociocultural contract theory, "good" (which is a moral term) has no objective, intercultural definition. So, one person in one culture may answer the question "yes" (basing his answer on the "goodness" of general behavior), and another person in another culture may answer the question "no" (same basis). If God does not actually exist, this answer does not change even if someone believes that He exists.
If an agnostic answers "without believing in God (whether God exists or not)," then the answer is a bit more complicated. We have to consider two possibilities here. The first being the atheistic position (described and answered above- which would hold true whether one believes or not) and the theistic position. "If God does exist and someone does not believe in Him, can that person still be good?" The answer is "yes," but with a qualifier.
First, let me establish the "yes" part. Theists believe that morals exist in God's nature. God cannot act immorally because it is not part of his nature. I am European by nature; therefore, I cannot be African- I perform actions that others might interpret as "African" (loosely- that was well established the last time I attempted to play Dance Central on the Xbox Kinect), but it is not truly "African", my acting is based on and performed in the context of my European nature. In the same way, God cannot be or do actual evil, because it is against his nature. He may do things that we interpret as being "evil" or "bad", but that is because we are not aware of all the factors affecting the situation (see my post "Suffering Sucks...or Does It?"), which is the greater context of God's all-loving and moral nature. Since we are assuming that God exists in this situation, "good" has an objective definition. People may choose to act "good" whether they believe in God or not. So, yes, a person may still "be good" if they do not believe in God, but only if God exists.
Now for the caveat. One may hold wrong beliefs, yet still choose to act in ways that their beliefs do not logically lead them to (see the Psychology Class Series). An example of this would be the atheist who wants to act benevolently and call it that. This is accomplished by the emotions. In the theistic worldview, man was created "in the image of God." Part of this "image" is possessing innate knowledge and understanding of the moral nature of God. The emotions react accordingly to this innate moral understanding and stimulate the person to act according to it rather than the according to the false beliefs. A person may still refuse to act according to the innate understanding of the moral law. This happens by a stronger emotion that is committed to the false belief (the result of the original "sin", according to the Christian worldview).
If a person chooses to act right even though they hold a view that does not allow for the right action, consistency is sacrificed in the worldview. So, yes a person may act good and do good things even though they don't believe that God exists, but that is only possible if God actually does exists. If God does not actually exist, then the question is nonsensical because one of the key terms is void of objective meaning ("good").