Fused filament fabrication (FFF) is a new procedure for the production of plastic parts, particularly if the parts have a complex geometry and are only needed in a limited quantity, e.g., in specific medical applications. In addition to the production of parts which are purely composed of polymers, fused filament fabrication can be successfully applied for the preparation of green bodies for sintering of metallic implant materials in medical applications. In this case, highly filled polymer–metal feedstocks, which contain a variety of polymeric components, are used. In this study, we focus on various polymer-metal feedstocks, investigate the rheological properties of these materials, and relate them to our results of FFF experiments. Small amplitudes of shear oscillations reveal that the linear range of the polymer–metal feedstocks under investigation is very small, which is caused by elastic and viscous interactions between the metallic particles. These interactions strongly influence or even dominate the flow properties of the feedstock depending on the applied shear stress. The magnitude of the complex viscosity strongly increases with decreasing angular frequency, which indicates the existence of an apparent yield stress. The viscosity increase caused by the high powder loading needed for sintering limits the maximum printing velocity and the minimum layer height. The apparent yield stress hinders the formation of smooth surfaces in the FFF process and slows down the welding of deposited layers. The influence of composition on the processing parameters (suitable temperature range) and part properties (e.g., surface roughness) is discussed on the basis of rheological data.