This letter considers the blind separation of convolutive mixtures in a multi-in-multi-out (MIMO) communication system. Multiple source signals are transmitted simultaneously over a shared communication medium (modeled as linear convolutive channels) to multiple receivers. We recast the joint recovery of the source signals, and the channel impulse responses as a block-rank-one matrix recovery problem, which can be effectively solved using a convex program. Our numerical experiments show that the proposed convex program yields exact recovery of the source signals when they are members of known generic subspaces. Moreover, the numerics also show that the successful recovery is achieved in blind MIMO with the number of measurements that scale roughly optimally with the number of unknowns. We discuss our results in the context of an application in blind MIMO wireless communications, where coding the transmitted messages results in exact blind separation at the receiver array.