The intrinsic viscosities for prolate and oblate spheroidal suspensions in a dilute Newtonian fluid are studied using a three-dimensional lattice Boltzmann method. Through directly calculated viscous dissipation, the minimum and maximum intrinsic viscosities and the period of the tumbling state all agree well with the analytical solution for particles with different aspect ratios. This numerical test verifies the analysis on maximum and minimum intrinsic viscosities. Different behavior patterns of transient intrinsic viscosity in a period are analyzed in detail. A phase lag between the transient intrinsic viscosity and the orientation of the particle at finite Reynolds number (Re) is found and attributed to fluid and particle inertia. At lower Re, the phase lag increases with Re. There exists a critical Reynolds number Rea at which the phase lag begins to decrease with Re. The Rea depends on the aspect ratio of the particle. We found that both the intrinsic viscosity and the period change linearly with Re when Re<Rea (low-Re regime) and nonlinearly when Re>Rea (high-Re regime). In the high-Re regime, the dependence of the period on Re is consistent with a scaling law, and the dependence of the intrinsic viscosity on Re is well described by second-degree polynomial fits.