This article presents a finite-volume scheme for multidimensional incompressible flows. Unstructured, solution-adaptive meshes composed of arbitrary convex polyhedra are used. A cell-centered equal-order formulation is developed. Gradients required for the evaluation of diffusion fluxes and for second-order-accurate convective operators are found by linear reconstruction. An additive-correction multigrid scheme is used to solve the resulting discrete equations. Pressure and velocity are stored at cell centers; momentum interpolation is used to prevent pressure checkerboarding. The SIMPLE algorithm is used for pressure-velocity coupling. Schemes for hanging-node and conformal adaption are implemented. The scheme is applied to benchmark problems using a variety of quadrilateral/hexahedral, triangular/tetrahedral, and hybrid meshes, and is shown to perform satisfactorily.