In this work, the effect of rigid rods on the discontinuous shear thickening (DST) transition in mixtures of smaller isotropic-shaped (calcium carbonate-CC) and larger rod-like (polyamide or glass) particles dispersed in water is experimentally established. The CC suspension in water is considered as a shear thickening matrix filling the pores of the fiber network. The DST in the matrix originates from the competition between the applied shear stress and the steric repulsion between adsorbed superplasticizer molecules. It is characterized by a typical S-shape of the flow curves, irregular oscillations of the shear rate in response to the applied shear stress, and some thixotropy. An addition of rods shifts the DST transition to lower critical shear rates, as explained by an increase in the suspension viscosity such that the shear rate to reach the onset stress of DST decreases. This behavior is satisfactorily reproduced by the reduced shear rate approach of Ohl and Gleissle [J. Rheol. 37, 381-406 (1993)] and, to a lesser extent, by the homogenization approach of Chateau et al. [J. Rheol. 52, 489-506 (2008)], both assuming random rod orientation. At fiber volume fractions, phi f >= 0.04, the mixture undergoes jamming, which is likely associated with the percolation threshold of the fiber network, nearly independent of the CC particle concentration. This idea is qualitatively supported by a modified homogenization approach, assuming that viscous dissipation mostly occurs in the vicinity of the contacts between fibers. The results of this work are believed to be useful for optimal formulations of fiber-reinforced cementitious materials.