We have studied avalanchelike rearrangements of domain patterns in the two-dimensional Ising model with static impurities, which is quenched to low temperatures. When breaking the up-down symmetry of the spins by a small applied field, the mere fluctuation of a single spin eventually results in a cascade of spin flips at the domain boundaries. We have analyzed the lifetime and size distribution functions for the avalanches and related the results to the general phenomena of self-organized criticality and to recent experiments on cellular magnetic domain patterns in magnetic garnet films. Our results suggest that the self-organized state in this system appears to be subcritical, in agreement with a recent theory.