The dishwasher is a nice idea, but I was never really all that trusting of them. Firstly, coming from a apartment maintenance man history, they can be quite crusty. They don't get rid of all the gunk they blast off the dishes. It tends to sit down in the screen or the pump. I would just soak them in oxyclean to get rid of the biologics and the labels and then give them a rinse in your sanitizer.
The dishwasher is nice for cleaning of big stuff, but getting into all the nooks of a bottle I wouldn't trust it.