The capability of the lattice Boltzmann method as an accurate mesoscopic solver for unsteady non-Newtonian flows is shown by investigating pulsatile shear-thinning blood flow in a three-dimensional idealised vessel. The non-Newtonian behaviour of blood flow is modelled by the Carreau-Yasuda viscosity model. Higher velocity and shear stress magnitudes, relative to Newtonian cases, are observed for the shear-thinning simulations in response to changes in the shear-rate dependent Womersley parameter. We also investigate the flexibility of the method through the shear-thinning behaviour of the lattice Boltzmann relaxation parameter at different Deborah numbers.