We investigate the role and behaviour of dust grains in fast C-type magnetohydrodynamic (MHD) shock waves in weakly ionized, dense molecular clouds. We calculate the structure of steady, oblique, C-type shocks with shock speed vs = 18 km s-1, propagating in a medium with number density nH = 105 cm, and magnetic field 0.3 mG. The angle between the pre-shock magnetic field and the shock normal is varied from 90° to ∼40° when the shocks become J-type. The grain population is represented by either a single-grain size or an MRN (Mathis, Rumpl and Nordsieck) grain size distribution with and with out polycyclic aromatic hydrocarbons (PAHs). The grain charge is assumed to vary, consistent with an approximate electron temperature profile within the shock. Grain inertia is neglected. Charged particle drifts, fluid velocities and the magnetic field are all permitted to have components perpendicular to the 'shock plane' containing the pre-shock (or post-shock) magnetic field and the shock normal. For the shock parameters considered here, small grains remain coupled to the magnetic field, while large grains are partially decoupled by collisions with the neutrals. The increase in grain charge within the shock front, due to the sticking of electrons, increases the magnetic coupling of the large grains, which acts to suppress the magnetic field and neutral velocity rotation out of the shock plane. Increasing the grain size increases the grain-neutral collisional heating leading to hotter, thinner shocks. The presence of PAHs reduces the electron abundance and grain charging is insignificant; the grain coupling to the magnetic field is therefore reduced and the rotation of the magnetic field out of the shock plane increases. Even in this case, the rotation is less than 5°. The primary effect of including the charged particle drift components perpendicular to the shock plane is to increase the dissipation rate and reduce the shock thickness.