We report measurements and analyses of resistivity, thermopower, and thermal conductivity of polycrystalline samples of perovskite LaRh$_{1-x}$Ni$_x$O$_3$. The thermopower is found to be large at 800 K (185 $mu$V/K for $x=$0.3), which is ascribed to the high-temperature stability of the low-spin state of Rh$^{3+}$/Rh$^{4+}$ ions. This clearly contrasts with the thermopower of the isostructural oxide LaCoO$_3$, which rapidly decreases above 500 K owing to the spin-state transition. The spin state of the transition-metal ions is one of the most important parameters in oxide thermoelectrics.