By using secular moral arguments, we must face the real dilemmas head on and take full responsibility for the actions we finally decide.
But the funny thing is that, as you probably know, theists do this just like atheists - it's an illusion that they get their 'absolute' morality from the Bible when they themselves (or their pastors) pick and choose what passages to follow and how to interpret them.
For example, let's say you've got three societies. In Society
A, the inhabitants live in a right-wing free marker system and the passages from Jesus about selling everything you have and giving the money you gain to the poor is ignored completely or explained away (for example by saying that the order applied only to the person Jesus was speaking to, or only to Jesus' disciples). In Society
B, the passage is not ignored, but rather used as a reason for maintaining a socialist democracy with a high income tax, a significant part of which goes to the citizens of the lower class. And finally, in Society
C, the passage, and those similar to it in meaning, are interpreted litterally, and the citizens live in a Communist society with a plan economy and no private property.
Society
A, B, and
C are vastly different, but adhere to the same book, and all could change gradually over time, just like real-world Christian countries have and continue to do. If the citizens of Society
C, for example, figure out that their Communist system doesn't work, they may gradually move to the right towards a mixed economy system, and in the process reinterpret the passages on property and money. In fifty years, Society
C may interpret them as metaphors, and live as Society
B while still claiming that the Bible is the source of their economic philosophy.
Both theists and atheists determine right and wrong the same way - by having society as a whole slowly shape it into what the majority wants it to be. Morality is ever-changing, even among the most fanatical of religious societies.