We introduce a numerical algorithm for solving anisotropic eikonal equations, whose complexity is quasi-linear O(N ln^2(N/ε)) with respect to the number N of discretization points, and logarithmic w.r.t. the numerical tolerance ε>0, with explicit constants depending on the metric defining the PDE geometry. In contrast with the fast-marching method, our algorithm does not rely on the causality property, and for this reason it can be applied to a variety of discretization schemes: semi-Lagrangian, Eulerian, or based on a Lax-Friedrichs relaxation of the eikonal PDE. Our method uses a narrow band to compute the eikonal front propagation, whose width is tuned depending on the properties of the discretization scheme and of the metric. Numerical experiments, involving anisotropic metrics arising in seismology, image segmentation and motion planning, illustrate the efficiency of the method.