Two- and three-dimensional networks of a columnar type are described, which result from partitions based on Poisson point distributions. The metric and topological properties of such laminated Poisson networks are derived and the applicability of the Lewis and Aboav-Weaire laws to them is tested. The 2D Poisson network contains cells with i>or=4 (i is the number of sides) and both laws are obeyed. The 3D Poisson networks are of various types and have F>or=6 (F is the number of faces in a cell) and F=14. In one particular type of 3D Poisson network the two laws are again exactly obeyed. In another type, the laws show large deviations at low F but are asymptotically obeyed when F tends to infinite.