Epipolar plane images (EPIs) contain special linear structures that reflect the disparity of a 3D point and are widely used in light field depth estimation. However, previous EPI-based approaches only utilize horizontal and vertical EPIs to estimate local disparities and ignore diagonal directions. In order to make full use of the regular grid light field images, we develop a strategy to extract epipolar plane images in all available directions. Based on the multi-orientation EPIs, a specific EPI in which the point is not occluded is found and used to calculate robust depth estimation. We also design a novel framework to estimate the depth information which combines the local depth with edge orientation. The multi-orientation EPIs and optimal orientation selection are proved to be effective in detecting and excluding occlusions. Experimental results show that the proposed method outperforms state-of-the-art depth estimation methods, especially near occlusion boundaries.