Magnetic resonance diffusion tensor tractography is a powerful tool for the non-invasive depiction of the white matter architecture in the human brain. However, due to limitations in the underlying tensor model, the technique is often unable to reconstruct correct trajectories in heterogeneous fiber arrangements, such as axonal crossings. A novel tractography method based on fast marching (FM) is proposed which is capable of resolving fiber crossings and also permits trajectories to branch. It detects heterogeneous fiber arrangements by incorporating information from the entire diffusion tensor. The FM speed function is adapted to the local tensor characteristics, allowing in particular to maintain the front evolution direction in crossing situations. In addition, the FM's discretization error is reduced by increasing the number of considered possible front evolution directions. The performance of the technique is demonstrated in artificial data and in the healthy human brain. Comparisons with standard FM tractography and conventional line propagation algorithms show that, in the presence of interfering structures, the proposed method is more accurate in reconstructing trajectories. The in vivo results illustrate that the elucidated major white matter pathways are consistent with known anatomy and that multiple crossings and tract branching are handled correctly.