Funnily enough, we just hit a similar issue.
I'm part of an environmental group at my workplace, and one of our recent pilots was to get rid of single-use paper cups. It was crazy: despite ample clean ceramic cups being available --and indeed stacked on top of the coffee machine!-- many people defaulted to paper cups, sometimes to the tune of having 5-10 used paper cups on their desk at the end of a day.
Anyway, one of the issues that the pilot uncovered was that no-one was loading or unloading the dishwasher: it transpired that everyone thought it was someone else's job. This led to piles of dirty cups and a shortage of clean cups, despite each kitchen area having a dedicated commercial-grade dishwasher.
The long-term solution is education - so that people understand that we must all take responsibility for the cleaning together, and to train people how to use the dishwashers. The short-term solution is for each kitchen area to have a couple of colleagues who keep an eye on things.
I'm one of them for my floor, and I unexpectedly find it strangely therapeutic to do the dish-washing :)