We study suspensions of oblate rigid particles in a viscous fluid for different values of the particle volume fractions. Direct numerical simulations have been performed using a direct-forcing immersed boundary method to account for the dispersed phase, combined with a soft-sphere collision model and lubrication corrections for short-range particle-particle and particle-wall interactions. With respect to the single-phase flow, we show that in flows laden with oblate spheroids the drag is reduced and the turbulent fluctuations attenuated. In particular, the turbulence activity decreases to lower values than those obtained by accounting only for the effective suspension viscosity. To explain the observed drag reduction, we consider the particle dynamics and the interactions of the particles with the turbulent velocity field and show that the particle-wall layer, previously observed and found to be responsible for the increased dissipation in suspensions of spheres, disappears in the case of oblate particles. These rotate significantly slower than spheres near the wall and tend to stay with their major axes parallel to the wall, which leads to a decrease of the Reynolds stresses and turbulence production and so to the overall drag reduction.