Cardiac tissue conductivity measurements can be used to assess the electrical substrate underlying normal and abnormal wavefront propagation. We describe a method of solving the inverse cardiac bidomain model to estimate average longitudinal and transverse intra and extra-cellular conductivities and fiber angle relative to an electrode array placed arbitrarily on the epi- or endocardial surface. A Newton-Raphson reconstruction method and two Tikhonov-type regularizations were able to stably identify conductivities and fiber angles in tissue models having anisotropies similar to those in real cardiac tissue. The reconstruction methods were tested with data from increasingly realistic two dimensional cardiac bidomain models and performed well both when measurement noise was added, and when simulated experimental and forward model matching was diminished. This approach may be a suitable basis for continuous monitoring of myocardial condition in-vivo via a catheter based electrode array.