Fused Filament Fabrication (FFF) is a widely used Additive Manufacturing technology with a growing user base. Unfortunately, users of FFF have limited resources for easily and cost-efficiently characterizing the materials from the ever-expanding polymer filament market. To address this issue, we constructed a low-cost device and measuring procedure based on the pressure drop in the polymer melt flow through an FFF nozzle which is used to compute the filament material rheological properties. In this approach, an analytical pressure drop model based on the Generalized-Newtonian-Fluid power-law model is employed where the consistency and power-law indices are determined from the measured pressure drop and volumetric flow rate through a nonlinear regression curve fit. The applicability of our device was demonstrated with six commercially available FFF filaments (four unfilled polymers and two carbon fiber polymer composites) by comparing the predicted viscosity model parameters to those obtained from measurements using a commercial rotational rheometer. A Filament Flow Index (FFI) capable of characterizing the flow of melted filament similar to Melt Flow Index is also proposed to assess the extrusion of materials used in FFF part fabrication.