We tackle the challenging problem of efficient and accurate seismic traveltime computation in 3D anisotropic media by applying the fast-sweeping method to a discontinuous Galerkin (DG)-based eikonal solver. Using this method leads to a stable and highly accurate scheme, which is faster than finite-difference schemes for a given precision, and with a low computational cost compared to the standard Runge-Kutta DG formulation. The integral formulation of the DG method also makes it easy to handle seismic anisotropy and complex topographies. Several numerical tests on complex models, such as the 3D SEG advanced modeling model, are given as illustration, highlighting the efficiency and the accuracy of this new approach. In the near future, these results will be used together with accurate solvers for seismic amplitude and take-off angle computation to revisit asymptotic inversion (traveltime/slope tomography) and imaging approaches (quantitative migration involving amplitudes and angles).