Particle stress is known to play a vital role in turbulence modulation. However, the earlier studies on particle stress were mostly confined to the cases of spherical and elongated particles. In the present study, we investigated the stress generated in dilute suspensions of inertialess spheroids with various shapes in wall-bounded turbulence. We performed direct numerical simulations of turbulent particle-laden channel flows utilizing a one-way coupled Eulerian-Lagrangian approach. The stress in the suspension of oblate spheroids was examined in detail and compared with prolate spheroid cases for the first time. The results show that particle stress is strongly dependent on particle shape and the stress of the oblate spheroid is qualitatively different from the case of prolate particles. However, we found that the fluctuating spanwise shear stress of flat oblate spheroids, which makes a significant contribution to turbulence dissipation, is in the same order of magnitude of the term generated by elongated prolate spheroids at a constant volume fraction. We also examined the effect of the Reynolds number on the particle stress in channel flows at Reτ = 180 and 1000. The results reveal a negligible influence on the mean stresses, but the fluctuating stresses are significantly Reynolds number dependent. In the buffer layer, we observed the correlations between the particle stress of spheroids and the fluctuating velocity of the fluid in the streamwise direction, which could be attributed to the different orientation of spheroids and fluid strain rate in low- and high-speed streaks of near-wall turbulence.