We present results of the masses of taste non-Goldstone $(F \ne \xi_5)$ pions and kaons calculated up to the next-to-leading order in the SU(3) staggered chiral perturbation theory (SChPT). The results can be used to fit data and to understand taste symmetry breaking effect quantitatively. The final expressions for the non-Goldstone masses contain 20 low energy constants unique to the non-Goldstone sector. We have calculated the several cases such as the full QCD, partially quenched QCD, and quenched QCD in the $N_f=1+1+1$ flavor and $N_f=2+1$ flavor cases in the SU(3) and SU(2) SChPT. In this paper, we present only the SU(3) part.