The ventricle is a strongly anisotropic three-dimensional excitable medium. Waves of electrical activity propagate faster parallel to the long axis of the muscle fibers. Moreover, this axis rotates intramurally across the ventricular wall. It is demonstrated that this rotating anisotropy can cause a transmural scroll vortex filament to spontaneously decay into wave turbulence above a minimum wall thickness comparable to the one necessary to sustain ventricular fibrillation. Moreover, the instability that produces this decay is shown to be associated with the propagation of localized twist-induced disturbances along the filament.