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We study the behaviour of confined cubic blue phases under shear flow via lattice Boltzmann simulations. We focus on the two experimentally observed phases, blue phase I and blue phase II. The disclination network of blue phase II continuously breaks and reforms under shear, leading to an oscillatory stress response in time. The oscillations are only regular for very thin samples. For thicker samples, the shear leads to a 'stick-slip' motion of part of the network along the vorticity direction. Blue phase I responds very differently: its defect network undergoes seemingly chaotic rearrangements under shear, irrespective of system size.