The two-dimensional fluidity of lipid bilayers enables the motion of membrane-bound macromolecules and is therefore crucial to biological function. Microrheological methods that measure fluid viscosity via the translational diffusion of tracer particles are challenging to apply and interpret for membranes, due to uncertainty about the local environment of the tracers. Here, we demonstrate a new technique in which determination of both the rotational and translational diffusion coefficients of membrane-linked particles enables quantification of viscosity, measurement of the effective radii of the tracers, and assessment of theoretical models of membrane hydrodynamics. Surprisingly, we find a wide distribution of effective tracer radii, presumably due to a variable number of lipids linked to each tracer particle. Furthermore, we show for the first time that a protein involved in generating membrane curvature, the vesicle trafficking protein Sar1p, dramatically increases membrane viscosity. Using the rheological method presented here, therefore, we are able to reveal a class of previously unknown couplings between protein activity and membrane mechanics.