This paper focuses on the development and application of a three-dimensional gas-kinetic Bhatnagar-Gross-Krook (BGK) method for the viscous flows in rotating machinery. For such flows, a rotating frame of reference is usually used in formulating the Navier-Stokes (N-S) equations, and there are two major concerns in constructing the corresponding BGK model. One is the change of the convective velocities in the N-S equations, which can be reflected through modification of the gas streaming velocity. The other one is the necessity to account for the effect of the additional Coriolis and centrifugal forces. Here, a specifically-designed acceleration term is added into the modified Boltzmann equation so that the source effects can be naturally included into the gas evolution process and the resulted fluxes. Under the finite-volume framework, the constructed BGK model is locally solved at each cell interface and then the numerical fluxes can be evaluated. When employing the BGK scheme, it is sometimes found that the calculated spatial derivatives of the initial and equilibrium distribution functions are sensitive to the mesh quality especially in complex rotating flow applications, which may significantly influence flux evaluation. Therefore, an improved approach for computing these slopes is adopted, through which the modeling capability for viscous flows is enhanced. For validation, several numerical examples are presented. The computed results show that the present method can be well applied to a wide range of flows in rotating machinery with favorable accuracy.