Conventional solutions of elastic wave equations rely on inaccurate finite-difference approximations of the time derivative, which result in strict dispersion and stability conditions and limitations. In this work, we derive a general solution to the elastic anisotropic wave equation, in the form of a Fourier Integral Operator (FIO). The proposed method is a generalization of the previously developed recursive integral time extrapolation operators from acoustic to elastic media, and can accurately propagate waves in time using the form of the analytical solution in homogeneous media. The formulation is closely connected to elastic wave mode decomposition, and can be applied to the most general anisotropic medium. The numerical calculation of the FIO makes use of a low-rank approximation to enable highly accurate and stable wave extrapolations. We present numerical examples including wave propagation in 3D heterogeneous orthorhombic and triclinic models.