Recent stellarator optimization efforts have targeted transport measures such as quasi-symmetry, effective ripple, and alignment of particle guiding center orbits with flux surfaces. This has resulted in significant reductions in neoclassical losses so that, at least for near-term experiments, the neoclassical transport of particles and energy can be made small compared to anomalous transport. However, momentum transport properties within magnetic flux surfaces provide an additional dimension for characterizing optimized stellarators. The momentum and flow damping features of optimized stellarators can vary widely, depending on their magnetic structure, ranging from systems with near-tokamak-like properties where toroidal flows dominate to those in which poloidal flows dominate and toroidal flows are suppressed. A set of tools has been developed for self-consistently evaluating the flow characteristics of different stellarators. Application of this model to existing and planned devices indicates that plasma flow properties vary significantly. Comparisons across devices can aid in unfolding the interplay between anomalous and neoclassical damping effects as well as the impact of momentum transport properties on related plasma phenomena such as turbulence suppression, shielding of resonant magnetic error fields, and impurity transport.