We present a numerical method for computing the first arrival time of seismic waves in media defined by a general Hooke tensor, in contrast with previous methods which are limited to a specific subclass such as the "tilted transversally isotropic" (TTI) model or "tilted orthorhombic" (TOR) model [WYF15, LBBMV19]. Our method proceeds in a single pass over the discretized domain, similar to the Fast Marching method, whereas existing methods for these types of anisotropies require multiple iterations, similar to the Fast Sweeping method. We introduce a new source factorization model, allowing us to achieve third order accuracy in smooth test cases. We also validate our solver by comparing it with the solution of the elastic wave equation on a 3D medium with general anisotropy.