We consider Ising spin systems, equivalently lattice gases evolving under discrete- or continuous-time Markov processes, i.e., "stochastic cellular automata" or "interacting particle systems." We show that for certain spin-flip probabilities or rates and suitable initial states the expectation values of products of spin variables taken at equal or different times are nonnegative; they satisfy the same inequalities as the equal-time correlations of ferromagnetic systems in equilibrium (first Griffiths inequality). Extensions of FKG inequalities to time-displaced correlations are also discussed.