The effect of temporal disorder on systems with up-down Z_2 symmetry is studied. In particular, we analyze two well-known families of phase transitions—the Ising and the generalized voter universality classes—and scrutinize the consequences of placing them under fluctuating global conditions. We observe that variability of the control parameter induces in both classes “temporal Griffiths phases” (TGPs). These recently uncovered phases are analogous to standard Griffiths phases appearing in systems with quenched spatial disorder, but where the roles of space and time are exchanged. TGPs are characterized by broad regions in parameter space in which (i) mean first-passage times scale algebraically with system size, and (ii) the system response (e.g., susceptibility) diverges. Our results confirm that TGPs are quite robust and ubiquitous in the presence of temporal disorder. Possible applications of our results to examples in ecology are discussed.