A reformulation of the Transmission Line Matrix (TLM) method is presented to model non-dispersive anisotropic media. Two TLM-based solutions to solve this problem can already be found in the literature, each one with an interesting feature. One can be considered a more conceptual approach, close to the TLM fundamentals, which identifies each TLM in Maxwell's equations with a specific line. But this simplicity is achieved at the expense of an increase in the memory storage requirements of a general situation. The second existing solution is a more powerful and general formulation that avoids this increase in memory storage. However, it is based on signal processing techniques and considerably deviates from the original TLM method, which may complicate its dissemination in the scientific community. The reformulation presented in this work exploits the benefits of both methods. On the one hand, it maintains the direct and conceptual approach of the original TLM, which may help to better understand it, allowing for its future use and improvement by other authors. On the other hand, the proposal includes an optimized treatment of the signals stored at the stub lines in order to limit the requirement of memory storage to only one accumulative term per field component, as in the original TLM versions used for isotropic media. The good behavior of the proposed algorithm when applied to anisotropic media is shown by its application to different situations involving diagonal and off-diagonal tensor properties.