We study the propagators for a large class of gravity theories having a nonzero, metric-compatible torsion. The theories are derivable from a Lagrangian containing all possible invariants quadratic or less in the torsion and Riemann curvature tensors, except that invariants are dropped if they do not contribute to the propagator in the linearized limit. Therefore, the torsion in these theories is, in general, a propagating field rather than one which vanishes outside matter. We study the constraints imposed on the propagator by the requirement that the theory have no ghosts or tachyons. In particular, we find that the addition of a spin-2+ torsion multiplet does not remove the spin-2+ ghost contributed by higher-derivative terms (Riemann curvature-squared terms). We discuss the phenomenology of theories with propagating torsion. The torsion must couple to spins with coupling constants much smaller than the electromagnetic fine-structure constant, or the force between two macroscopic ferromagnets, due to torsion exchange, would be huge, far greater than the familiar magnetic force due to photon exchange. We briefly discuss the phenomenology of propagating torsion "potentials." Theories involving such potentials have been proposed recently by several authors.