The authors consider the generalization to the anisotropic case of the numerical algorithm by [B. Merriman, J. Bence and S. Osher, Diffusion generated motion by mean curvature, Computational Crystal Growers Workshop, J.E. Taylor, ed., Sel. Lectures Math., AMS, Providence, RI, 1992, 73–83 (1992), see Zbl 0776.65002 for the entire collection] for the mean curvature flow. Let \(E\subset \mathbb{R}^n\) be a closed set with compact boundary, and the anisotropy \( (\phi,\phi^0)\) a pair of mutually polar, convex, 1-homogeneous functions on \(\mathbb{R}^n\). Define \(T_h(E)=\{x:u(x,h)\geq 1/2 \}\), where \(u\) solves: \[ \begin{cases}{\partial u \over \partial t}(x,t)\in \text{div}(\phi^0(\nabla u)\partial \phi^0(\nabla u))(x,t), \;t>0, \;x\in \mathbb{R}^n,\cr u (\cdot ,0)=\chi_E.\end{cases} \] Their main result is a consistency theorem: if there exists a “regular” evolution starting from \(E\), then \(T_h^{[t/h]}(E)\) converges to this evolution as \(h \rightarrow 0\) ( \([t/h]\) is the integer part of \(t/h\)). Such evolutions are a variant of those defined in [G. Bellettini, M. Novaga, Math. Models Methods Appl. Sci. 10, 1–10 (2000; Zbl 1016.53048)], and exist e.g. under various smoothness and convexity assumptions on \(\phi, \phi^0, E\), and \(\partial E\). This consistency result, together with the monotonicity of the scheme (\(E \subseteq F \Rightarrow T_h E \subseteq T_hF\dots\)) yields convergence also to all generalized solutions defined (in the smooth case) using barriers as long as these are unique. Also, it yields the convergence of the scheme to crystalline evolutions, when the initial set is convex. The proof relies on the construction of suitable sub and supersolutions.