Tensor eigenproblems have wide applications in blind source separation, magnetic resonance imaging, and molecular conformation. In this study, we explore an alternating direction method for computing the largest or smallest Z-eigenvalue and corresponding eigenvector of an even-order symmetric tensor. The method decomposes a tensor Z-eigenproblem into a series of matrix eigenproblems that can be readily solved using off-the-shelf matrix eigenvalue algorithms. Our numerical results show that, in most cases, the proposed method converges over two times faster and could determine extreme Z-eigenvalues with 20–50% higher probability than a classical power method-based approach.