It has long been known that strings wound around incontractible cycles can play a vital role in cosmology. In particular, in a spacetime with toroidal spatial hypersurfaces, the dynamics of the winding modes may help yield three large spatial dimensions. However, toroidal compactifications are phenomenologically unrealistic. In this paper we therefore take a first step toward extending these cosmological considerations to D-dimensional toroidal orbifolds. We use numerical simulation to study the time scales over which "pseudo-wound" strings unwind on these orbifolds with trivial fundamental group. We show that pseudo-wound strings can persist for many "Hubble times" in some of these spaces, suggesting that they may affect the dynamics in the same way as genuinely wound strings. We also outline some possible extensions that include higher-dimensional wrapped branes.