We present diffusing-wave spectroscopy measurements of bubble dynamics in a continuously sheared aqueous foam. At slow strain rates, isolated clusters of bubbles intermittently rearrange from one solidly packed configuration to another, even though the macroscopic flow appears continuous. At fast rates, bubbles instead move smoothly and continuously throughout the entire foam. In other words, shear flow that appears macroscopically laminar is similarly laminar down to the bubble scale; effectively the foam "melts." The crossover to this regime can be understood in terms of elastic energy accumulation and viscous dissipation mechanisms. In particular, the strain rate needed for shear-induced melting to occur is set by the ratio of a yield strain to the rearrangement event duration. To explore the implications for macroscopic flow, we compare these bubble-scale dynamics directly with viscosity measurements.