The purpose of this paper is to present a numerical methodology for the computation of complex 3D turbomachinery flows using advanced multiequation turbulence closures, including full seven-equation Reynolds-stress transport models. The flow equations are discretized on structured multiblock grids, using an upwind biased (MUSCL reconstruction) finite-volume scheme. Time integration uses a local dual-time-stepping implicit procedure, with internal subiterations. Computational efficiency is achieved by a specific approximate factorization of the implicit subiterations, designed to minimize the computational cost of the turbulence transport equations. Convergence is still accelerated using a mean-flow-multigrid full-approximation-scheme method, where multigrid is applied only on the mean-flow variables. Speed-ups of a factor 3 are obtained using three levels of multigrid (fine plus two coarser grids). Computational examples are presented using two Reynolds-stress models, and also a baseline model, for various turbomachinery configurations, and compared to available experimental measurements.