Two classes of cyclic, successive first-order linear, irreversible chemical reactions are found to exhibit chemical oscillations, and we elucidate how they show up. Furthermore, these are applied to stationary open systems where a starting chemical species is pumped in as a sinuasoidal function of time and an intermediate species is pumped out as the same function, and spectra for the rates of participating species are obtained which show sharp resonance peaks that are missing in nonoscillating systems. Hence, the present cyclic model suggests that even such simple reactions can undergo resonance motion which can produce strong harmonic activities similar to the usual ones of under damped harmonic oscillators.