In this paper, we present a computationally efficient semi-implicit scheme for the simulation of three-dimensional hydrostatic free surface flow problems on staggered unstructured Voronoi meshes. For each polygonal control volume, the pressure is defined in the cell center, whereas the discrete velocity field is given by the normal velocity component at the cell faces. A piecewise high-order polynomial vector velocity field is then reconstructed from the scalar normal velocities at the cell faces by using a new high-order constrained least-squares reconstruction operator. The reconstructed high-order piecewise polynomial velocity field is used for trajectory integration in a semi-Lagrangian approach to discretize the nonlinear convective terms in the governing PDE. For that purpose, a high-order Taylor method is used as ODE integrator. The resulting semi-implicit algorithm is extensively validated on a large set of different academic test problems with exact analytical solution and is finally applied to a real-world engineering problem consisting of a curved channel upstream of two micro-turbines of a hydroelectric power plant. For this realistic case, some experimental reference data are available from field measurements. Copyright © 2012 John Wiley & Sons, Ltd.