Dynamics of fracture is investigated in an anisotropic two-dimensional Born-Maxwell model by numerical simulations. From previous studies it is known that the isotropic model shows crack branching and velocity oscillations of the propagating main crack above a critical velocity, similarly with experimental findings in some brittle materials. Here we present studies in which anisotropy has been introduced to the model system. Anisotropy is found to have significant effects on crack propagation and on the pattern it forms. In the case of symmetric anisotropy (relative to the crack direction) we found changes in velocity oscillation and side branching properties. In the case of asymmetric anisotropy two kinds of periodicities occur and strong anisotropy causes different branch patterns to form at two sides of the main crack. In addition, the role of disorder through distributed spring constants has been studied for both types of anisotropy. Finally a simple exactly solvable model for investigating the initial stages of crack branching has been developed and analyzed.