We devise and analyze vertex-based schemes on polyhedral meshes to approximate advection-reaction equations. Error estimates of order 3/2 in the discrete inf-sup stability norm are established. The two key ingredients are a local poly-hedral reconstruction map leaving affine polynomials invariant, and a local design of stabilization whereby gradient jumps are only penalized across some subfaces in the interior of each mesh cell. Numerical results are presented on three-dimensional polyhedral meshes.