After introducing a class of string-inspired unconventional cosmologies, I will discuss the spectra of adiabatic scalar and tensor perturbations therein. On the basis of a toy model I will argue that those spectra are quite likely too "blue" to be responsible for the observed large-scale structure of our Universe. I will then discuss a possible way out of this impasse whereby a scale-invariant spectrum of adiabatic density fluctuations emerges through the early decay of the Kalb-Ramond axion of string theory.