We consider a general robust block-structured optimization problem, coming from applications in network and energy optimization. We propose and study an iterative cutting-plane algorithm, generic for a large class of uncertainty sets, able to tackle large-scale instances by leveraging on their specific structure. This algorithm combines known techniques (cutting-planes, proximal stabilizations, efficient heuristics, warm-started bundle methods) in an original way for better practical efficiency. We provide a theoretical analysis of the algorithm and connections to existing literature. We present numerical illustrations on real-life problems of electricity generation under uncertainty. These clearly show the advantage of the proposed regularized algorithm over classic cutting plane approaches. We therefore advocate that regularized cutting plane methods deserve more attention in robust optimization. © 2017, Springer-Verlag Berlin Heidelberg.