The stability of the gyroid phase of diblock copolymers has been studied using combined oscillatory shear and small-angle neutron scattering (SANS) techniques. It is shown that the gyroid phase of polystyrene-polyisoprene (PS-PI) is unstable when exposed to combined large-amplitude and high-frequency shear deformations. The bicontinuous cubic gyroid structure (G) transforms to the hexagonally cylinder phase (HEX). The transition is perfectly reversible, but with a significant difference in time constants. Upon application of shear the G --> HEX transition is instantaneous within experimental resolution, while the HEX --> G relaxation after cessation of shear takes hours. The texture of the shear-induced cylinder phase is shown to be a near ideal monodomain, while the relaxed gyroid phase constitutes a two-dimensional powder with the characteristic 10-spot scattering pattern. The shear-induced destabilization is discussed in relation to analogous observations on shear-induced order-to-order and disorder-to-order transitions observed in related block copolymer systems and in microemulsions. It is discussed whether these phenomena originate in shear-reduced fluctuations or shear-induced dislocations.