The first full-scale three-dimensional core-collapse supernova (SN) simulations with sophisticated neutrino transport show pronounced effects of the standing accretion shock instability (SASI) for two high-mass progenitors (20 and 27 M([Symbol: see text])). In a low-mass progenitor (11.2 M([Symbol: see text])), large-scale convection is the dominant nonradial hydrodynamic instability in the postshock accretion layer. The SASI-associated modulation of the neutrino signal (80 Hz in our two examples) will be clearly detectable in IceCube or the future Hyper-Kamiokande detector, depending on progenitor properties, distance, and observer location relative to the main SASI sloshing direction. The neutrino signal from the next galactic SN can, therefore, diagnose the nature of the hydrodynamic instability.