  In this paper, we propose a new scheme for anisotropic motion by mean
curvature in $\R^d$. The scheme consists of a phase-field approximation of the
motion, where the nonlinear diffusive terms in the corresponding anisotropic
Allen-Cahn equation are linearized in the Fourier space. In real space, this
corresponds to the convolution with a kernel of the form \[ K_{\phi,t}(x) =
\F^{-1}\left[ e^{-4\pi^2 t \phi^o(\xi)} \right](x). \] We analyse the resulting
scheme, following the work of Ishii-Pires-Souganidis on the convergence of the
Bence-Merriman-Osher algorithm for isotropic motion by mean curvature. The main
difficulty here, is that the kernel $K_{\phi,t}$ is not positive and that its
moments of order 2 are not in $L^1(\R^d)$. Still, we can show that in one sense
the scheme is consistent with the anisotropic mean curvature flow.
