Primal-dual proximal optimization methods have recently gained much interest for dealing with very large-scale data sets encoutered in many application fields such as machine learning, computer vision and inverse problems [1-3]. In this work, we propose a novel random block-coordinate version of such algorithms allowing us to solve a wide array of convex variational problems. One of the main advantages of the proposed algorithm is its ability to solve composite problems involving large-size matrices without requiring any inversion. In addition, the almost sure convergence to an optimal solution to the problem is guaranteed. We illustrate the good performance of our method on a mesh denoising application.