Every night in yet another house in Afghanistan's capital, a U.S. green card-holding couple from California take turns sleeping, with one always awake to watch over their three young children so they can flee if they hear the footsteps of the Taliban.

They've moved seven times in two weeks, relying on relatives to take them in and feed them. Their days are an uncomfortable mix of fear and boredom, restricted to a couple of rooms where they read, watch TV and play "The Telephone Game" in which they whisper secrets and pass them on, a diversion for the children that has the added benefit of keep...