Magnetic Resonance Electrical Impedance Tomography (MREIT) brings high resolution imaging of true conductivity distribution to reality. MREIT images are reconstructed based on measurements of current density distribution and a surface potential value, induced by an externally applied current flow. Since biological tissues may be anisotropic, isotropic conductivity assumption, as it is adopted in most of MREIT reconstruction algorithms, introduces reconstruction inaccuracy. In this study, a novel algorithm is proposed to reconstruct MREIT images of anisotropic conductivity. Relative values of anisotropic conductivity are reconstructed iteratively, using only measurement of current density distribution. By measuring a surface potential or a conductivity value, true values of anisotropic conductivity can be recovered. The technique is evaluated based on simulated measurements with and without additive noise. The results show that anisotropic and isotropic conductivity distributions can be reconstructed, successfully.