With falling ball rheometry, we have measured the apparent relative viscosity of suspensions of large, neutrally buoyant, rigid rods in a viscous Newtonian fluid, while approximately maintaining the rods in a randomly oriented configuration. A new technique for measuring the time of flight of a ball between two positions is used. This computerized technique, based upon an eddy current detector, enables us to determine the position of a metallic (nonmagnetic) ball falling through an opaque suspension, with high accuracy (less than 1.5% error). The rods for the suspensions had a nominal aspect ratio of 10 and experiments were carried out at a single volume fraction, 0.05. Two populations of rods were used to having nominal diameters of 1.5875 mm and 3.175 mm. To within the errors of these experiments, suspensions from both populations had the same relative viscosity, with the overall average being 1.457. This viscosity was significantly different from that of a similar suspension (volume fraction=0.05) of rods of nominal aspect ratio 20 and it agreed well with theoretical results for the viscosity of a dilute suspension of randomly oriented rods.