Numerical results, based on a lattice method for computational general relativity, will be presented for Cauchy evolution of initial data for the Brill, Teukolsky and polarized Gowdy spacetimes. The simple objective of this paper is to demonstrate that the lattice method can, at least for these spacetimes, match results obtained from contemporary methods. Some of the issues addressed in this paper include the handling of axisymmetric instabilities (in the Brill space-time) and an implementation of a Sommerfeld radiation condition for the Brill and Teukolsky spacetimes. It will be shown that the lattice method performs particularly well in regard to the passage of the waves through the outer boundary. Questions concerning multiple black holes, mesh refinement and long term stability will not be discussed here but may form the basis of future work.