We report detailed experimental studies on the rheology of cubic blue phases. We observe several flow regimes within each blue phase from rheomicroscopy and small angle light scattering experiments. Both the cubic blue phases exhibit solid-like response while the cholesteric phase shows gel-like behavior. The elastic modulus of BP-I is larger than that of BP-II. The shear induced yield transition occurs at a higher strain in BP-II than BP-I. Both the blue phases show stress relaxation through periodic modulation in step-strain experiments. Our results show that the rheological responses of these two phases are significantly different owing to the distinct networks of defect-disclinations.