I've come to the conclusion that the Problem of Evil is far too open to interpretation by believers to make them realize the actual logical problem involved. They can merely say that atheists do not have a definitive definition of "evil," which seems to be a sufficient rationalization for many. Christians tend to think of the argument as something attune to, "God lets bad things happen, therefore he doesn't exist," which simply isn't the case. It's meant to demonstrate an inherent contradiction in the Christian concept of God without any non-Christian definition of "evil" being necessary. So, I've decided to couch the argument in terms of the Christian concept of sin. Sin is generally what Christians mean when they talk of evil, and it is what non-believers mean as well when they use the Problem of Evil correctly. My argument utilizes the idea that sin is an action that is against God's will. That's not to say that that's all sin is, but that it's a fundamental aspect of what sin is. To say that X is a sin, it is to say that God disapproves of X, or that God wills or desires not-X.
My argument also assumes a concept of God that at least incorporates basic ideas of omnipotence, omniscience, and intentionality. By saying that God is omnipotent, I simply mean that He can at least actualize that which He wills and prevent the actualization of that which He does not will. By saying that God is omniscient, I simply mean that He can at least fully understand what will be actualized by his action or inaction, as well as what will be actualized by the action or inaction of beings that he created. By saying that God is intentional, I simply mean that God acts deliberately to bring about that which He wills, and prevent that which he does not will. And, by "Christianity," I mean a construct that at least incorporates this concept of God as well as the concept of sin mentioned above.
That being said, here's my argument:
- Sin is an action that is against God's will. [Premise]
- God can actualize what he wills, and can prevent the actualization of that which he does not will. (IOW, God is omnipotent) [Premise]
- God knows fully what will be actualized through His action or inaction, or through the action or inaction of a being that he created. (IOW, God is omniscient) [Premise]
- God seeks to actualize what he wills, and prevent what he does not will. (IOW, God acts intentionally) [Premise]
- Therefore, God will actualize that which he wills, and prevent that which he does not will. [2, 3, 4]
- Corollary: Everything that is actualized is what God wills. 
- If God exists, sin cannot exist. [1, 6]
- Christianity incorporates the concept of God and sin. [Premise]
- Therefore, Christianity is false. [7, 8]