Microstructural information over an entire sample is important to understand the macroscopic behavior of materials. X-ray scattering tensor tomography facilitates the investigation of the microstructural organisation in statistically large sample volumes. However, established acquisition protocols based on scanning small-angle X-ray scattering and X-ray grating interferometry inherently require long scan times even with highly brilliant X-ray sources. Recent developments in X-ray diffractive optics towards circular pattern arrays enable fast single-shot acquisition of the sample scattering properties with 2D omnidirectional sensitivity. Leveraging on these new optical elements, we propose here simple yet inherently rapid acquisition protocols forX-ray scattering tensor tomography. Results from both simulation and experimental data, supported by a null space analysis, suggest that the proposed acquisition protocols are not only rapid but also corroborate that sufficient information for the accurate volumetric reconstruction of the scattering properties is provided. The proposed acquisition protocols will build the basis for rapid inspection and/or time-resolved tensor tomography of the microstructural organisation over an extended field of view.