We use optical microscopy to investigate the yielding mechanism of concentrated emulsions submitted to an oscillatory strain. While rheological measurements suggest a smooth transition from linear elasticity to plastic behavior as the amplitude of the imposed strain is increased, direct imaging of the sheared emulsions reveals that at a microscopic level yielding occurs abruptly. Indeed, we find that a well-defined critical strain c separates a regime where no irreversible droplet rearrangements occur during oscillatory strain from a plastic regime where almost all droplets undergo irreversible displacements. Above the volume fraction at which droplets achieve random close packing, the amplitude of the critical strain c increases linearly with droplet volume fraction. Surprisingly, we find that yielding is associated with very small droplet displacements, with almost no structural rearrangement. These rearrangement events are very heterogeneous both in space and time, with bursts of plastic activity separating long quiescent periods. The typical size of the rearranged regions is modest at random close packing and increases with increasing droplet concentration.