Known to be structured in several patterns at the same time, the prior image of interest is always modeled with the idea of enforcing multiple constraints on unknown signals. For instance, when dealing with a hyperspectral restoration problem, the combination of constraints with piece-wise smoothness and low rank has yielded promising reconstruction results. In this paper, we propose a novel mixed-noise removal method by employing 3D anisotropic total variation and low rank constraints simultaneously for the problem of hyperspectral image (HSI) restoration. The main idea of the proposed method is based on the assumption that the spectra in an HSI lies in the same low rank subspace and both spatial and spectral domains exhibit the property of piecewise smoothness. The low rankness of an HSI is approximately exploited by the nuclear norm, while the spectral-spatial smoothness is explored using 3D anisotropic total variation (3DATV), which is defined as a combination of 2D spatial TV and 1D spectral TV of the HSI cube. Finally, the proposed restoration model is effectively solved by the alternating direction method of multipliers (ADMM). Experimental results of both simulated and real HSI datasets validate the superior performance of the proposed method in terms of quantitative assessment and visual quality.