Ultrasound as an acoustic imaging modality suffers from various kinds of noise. The presence of noise especially hinders the 3D visualization of ultrasound data, both in terms of resolving the spatial occlusion of the signal by surrounding noise, and mental decoupling of the signal from noise. This paper presents a novel type of structure-preserving filter that has been specifically designed to eliminate the presence of speckle and random noise in 3D ultrasound datasets. This filter is based on a local distribution of variance for a given voxel. The lowest variance direction is assumed to be aligned with the direction of the structure. A streamline integration over the lowest-variance vector field defines the filtered output value. The new filter is compared to other popular filtering approaches and its superiority is documented on several use cases. A case study where a clinician was delineating vascular structures of the liver from 3D visualizations further demonstrates the benefits of our approach compared to the state of the art.