This paper develops and applies a linear viscoelastic model for bending and torsional modes of fluid membranes, based on the nonlinear Cosserat surface fluid model. The linearized fluid membrane model in spherical and cylindrical geometries is shown to decouple bending and torsional viscoelastic modes. It is found that solutions of the membrane viscoelastic model to small-amplitude oscillatory bending and torsion allows for the measurement of the bending and the torsion viscosity. The model and its potential in characterizing the bending and torsion viscoelasticity of membranes complements the on-going efforts to establish the role of curvature in dissipative process of biological membranes.