In this paper, we propose an efficient matrix-free algorithm to reconstruct locations and size of flaws in a specimen from volumetric ultrasound data by means of a native 3D Sparse Signal Recovery scheme using Orthogonal Matching Pursuit (OMP). The efficiency of the proposed approach is achieved in two ways. First, we formulate the dictionary matrix as a block multilevel Toeplitz matrix to minimize redundancy and thus memory consumption. Second, we exploit this specific structure in the dictionary to speed up the correlation step in OMP, which is implemented matrix-free. We compare our method to state-of-the-art, namely 3D Synthetic Aperture Focusing Technique, and show that it delivers a visually comparable performance, while it gains the additional freedom to use further methods such as Compressed Sensing.