The response of fractal networks of dispersed single walled carbon nanotubes in poly(ethylene oxide) to continuous constant-rate shear flow is examined as functions of shear rate and nanotube concentration. The steady shear viscosity values are strong functions of shear rate and follow a power-law shear-thinning character, while the nanotube concentration dependence of the viscosity (at a fixed shear rate) is somewhat weaker than the scaling of the equilibrium modulus values. For dispersions with nanotube concentrations corresponding to the semidilute regime, the stress response to constant-rate continuous shear from rest demonstrates a stress maximum that decays to a steady value at long times. The stress maximum and steady shear behavior can be reconciled in terms of the changing structure at the mesoscale for the fractal networks. On the other hand, the transient development of the stress during the start-up experiments can be qualitatively reconciled in the context of a cluster dynamics model.