We explore the ability of directional nuclear-recoil detectors to constrain the local velocity distribution of weakly interacting massive particle ( WIMP ) dark matter by performing Bayesian parameter estimation on simulated recoil-event data sets . We discuss in detail how directional information , when combined with measurements of the recoil-energy spectrum , helps break degeneracies in the velocity-distribution parameters . We also consider the possibility that velocity structures such as cold tidal streams or a dark disk may also be present in addition to the Galactic halo . Assuming a \mathrm { CF } _ { 4 } detector with a 30-kg-yr exposure , a 50-GeV WIMP mass , and a WIMP-nucleon spin-dependent cross-section of 10 ^ { -3 } ~ { } \mathrm { pb } , we show that the properties of a cold tidal stream may be well constrained . However , measurement of the parameters of a dark-disk component with a low lag speed of \sim 50 ~ { } \mathrm { km / s } may be challenging unless energy thresholds are improved .