This paper continues in the previous research focussed to two simple questions. The first one reads: "What is the influence of anisotropy of computational lattice on simulations of boundary movement?" where grain boundary movement typically appears in simulations of grain boundary migration and static/dynamic recrystallization. The second question reads: "How is the computational anisotropy related to natural anisotropy of the material lattice itself?" This study is focussed on the influence of change of the computational algorithm and/or lattice on the grain boundary movement. Two algorithms, the majority rule and the simple modification of the Monte Carlo method for two different lattices - namely square and hexagonal one - are used.