I’ve not really even been that bothered by the problem of theodicy, so, unfortunately, I can’t make good use of what I imagine Egnor though would be the most attractive question for atheists. I think most discussions of evil/theodicy can be split into two separate categories: evil that befalls people and evil that is committed by people, so I’ll tackle this question in two parts.
Evil that happens to people
This category includes harm that befalls people as a result of natural accident (earthquake, illness, etc) and deliberate action by others. This is the ‘why bad things happen to good people’ question.
For an atheist the answer is easy: there is no mechanism by which we can escape the logical or physical consequences of the actions of others or of nature. If it begins to rain, we will get wet, regardless of whether we deserve it or how urgent it was to stay dry. There is no way to negotiate with a hurricane, and, although you may try to reason with an enemy soldier, once she fires her gun, there is no arguing with her bullet. As long as there is the possibility of naturally occurring harm and/or deliberately engineered harm by others and if we do not have the superhuman foresight and power that would be necessary to be able to avoid these occurrences, undeserved evils will occur.
I think this exact explanation is available to religious people as well. Even someone who believes in occasional miracles would be hard-pressed to argue for a world of constant intercession to protect us from the consequences of other people’s actions. You quickly end up undermining the entire principle of causality and with it, the possibility of reasoning about our own actions and making reasonable choices. (I discuss this in more detail in my answer to Why is there Regularity/Law in Nature). This line of argument works as well for theists as for atheists; C.S. Lewis makes it very well in Mere Christianity.
There is no reason for anyone, theist or atheist, to be surprised or betrayed by the fact that nature can do us harm. Given our limited knowledge of the scope of our actions, we should be similarly unsurprised that other people do us unintentional or unwitting harm. The only possible avenue of objection is to the fact that people do intentional harm to others. We may justifiably ask why that particular evil exists.
Part Two of this answer, addressing Evil Chosen by People is now up!