Bidirectionality effects can be a significant confounding factor when measuring hyperspectral reflectance data. The bidirectional reflectance distribution function (BRDF) can effectively characterize the reflectivity of surfaces to correct remote sensing measurements. However, measuring BRDFs can be time-consuming, especially when collecting Mueller matrix BRDF (mmBRDF) measurements of a surface via conventional goniometric techniques. In this paper, we present a system for collecting mmBRDF measurements using static optical fiber detectors that sample the hemisphere surrounding an object. The entrance to each fiber contains a polarization state analyzer configuration, allowing for the simultaneous acquisition of the Stokes vector intensity components at many altitudinal and azimuthal viewing positions. We describe the setup, calibration, and data processing used for this system and present its performance as applied to mmBRDF measurements of a ground glass diffuser.