Also, just to be clear, I'm not advocating the complete removal of events or messages from the design. All I'm saying is that the components themselves should neither send nor receive these messages. The systems, however, are free to communicate among themselves. As a matter of a fact, signaling would probably be the most efficient way to handle the achievement system, since it only needs to run occasionally. Whenever another system modifies component state that could affect achievement status, it sends a signal to the achievement system to wake it up. The achievement system runs once, awarding any achievements it detects, then goes back to sleep until it receives another "reevaluate" signal.

As a matter of a fact, signaling would probably be the most efficient way to handle the achievement system, since it only needs to run occasionally.