In this paper, we propose a compressed sensing scheme for volumetric synthetic aperture measurements in ultrasonic nondestructive testing. The compression is achieved by limiting the measurement to a subset of the Fourier coefficients of the full measurement data, where we also address the issue of a suitable hardware architecture for the task. We present a theoretic analysis for one of the proposed schemes in terms of the Restricted Isometry Property and derive a scaling law for the lower bound of the number of necessary measurements for stable and efficient recovery. We verify our approach with reconstructions from measurement data of a steel specimen that was compressed synthetically in software. As a side result, our approach yields a variant of the 3-D Synthetic Aperture Focusing Technique which can deal with compressed data.