Coupled-channel π K and η K scattering amplitudes are determined by studying the finite-volume energy spectra obtained from dynamical lattice QCD calculations. Using a large basis of interpolating operators, including both those resembling a qq̅ construction and those resembling a pair of mesons with relative momentum, a reliable excited-state spectrum can be obtained. Working at m_π=391 MeV, we find a gradual increase in the J^P=0^+ π K phase-shift which may be identified with a broad scalar

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... sonance that couples strongly to π K and weakly to η K. The low-energy behavior of this amplitude suggests a virtual bound-state that may be related to the κ resonance. A bound state with J^P=1^- is found very close to the π K threshold energy, whose coupling to the π K channel is compatible with that of the experimental K^(892). Evidence is found for a narrow resonance in J^P=2^+. Isospin--3/2 π K scattering is also studied and non-resonant phase-shifts spanning the whole elastic scattering region are obtained.