We present precise measurements of the growth rate of cosmic structure for the redshift range 0.1 <z <0.9, using redshift-space distortions in the galaxy power spectrum of the WiggleZ Dark Energy Survey. Our results, which have a precision of around 10 per cent in four independent redshift bins, are well fitted by a flat I? cold dark matter (I?CDM) cosmological model with matter density parameter I?m= 0.27. Our analysis hence indicates that this model provides a self-consistent description of the growth of cosmic structure through large-scale perturbations and the homogeneous cosmic expansion mapped by supernovae and baryon acoustic oscillations. We achieve robust results by systematically comparing our data with several different models of the quasi-linear growth of structure including empirical models, fitting formulae calibrated to N-body simulations, and perturbation theory techniques. We extract the first measurements of the power spectrum of the velocity divergence field, PI?I?(k), as a function of redshift (under the assumption that, where g is the galaxy overdensity field), and demonstrate that the WiggleZ galaxy-mass cross-correlation is consistent with a deterministic (rather than stochastic) scale-independent bias model for WiggleZ galaxies for scales k <0.3hMpc-1. Measurements of the cosmic growth rate from the WiggleZ Survey and other current and future observations offer a powerful test of the physical nature of dark energy that is complementary to distance-redshift measures such as supernovae and baryon acoustic oscillations.