We present a simple model of a dynamical system driven slowly by externally-imposed coherent noise. Although the system never becomes critical in the sense of possessing spatial correlations of arbitrarily long range, it does organize into a stationary state characterized by avalanches with a universal power-law size distribution. We explain the behavior of the model within a time-averaged approximation, and discuss its connection to the dynamics of earthquakes, the Gutenberg-Richter law, and to recent experiments on avalanches in rice piles.