We have observed 99 mid-infrared-bright, massive young stellar objects and compact HII regions drawn from the Red MSX source survey in the J=3-2 transition of ^12^CO and ^13^CO, using the James Clerk Maxwell Telescope. 89 targets are within 6kpc of the Sun, covering a representative range of luminosities and core masses. These constitute a relatively unbiased sample of bipolar molecular outflows associated with massive star formation. Of these, 59, 17 and 13 sources (66, 19 and 15 percent) are found to have outflows, show some evidence of outflow, and have no evidence of outflow, respectively. The time-dependent parameters of the high-velocity molecular flows are calculated using a spatially variable dynamic time-scale. The canonical correlations between the outflow parameters and source luminosity are recovered and shown to scale with those of low-mass sources. For coeval star formation, we find the scaling is consistent with all the protostars in an embedded cluster providing the outflow force, with massive stars up to ~30M_{sun}_ generating outflows. Taken at face value, the results support the model of a scaled-up version of the accretion-related outflow-generation mechanism associated with discs and jets in low-mass objects with time-averaged accretion rates of ~ 10^-3^M_{sun}_/yr on to the cores. However, we also suggest an alternative model, in which the molecular outflow dynamics are dominated by the entrained mass and are unrelated to the details of the acceleration mechanism. We find no evidence that outflows contribute significantly to the turbulent kinetic energy of the surrounding dense cores.