We study the properties of kinematically disturbed galaxies in the Sydney-AAO Multi-object Integral field spectrograph (SAMI) Galaxy Survey using a quantitative criterion, based on kinemetry (Krajnović et al.). The approach, similar to the application of kinemetry by Shapiro et al., uses ionized gas kinematics, probed by Ha emission. By this method, 23 ± 7 per cent of our 360-galaxy sub-sample of the SAMI Galaxy Survey are kinematically asymmetric. Visual classifications agree with our kinemetric results for 90 per cent of asymmetric and 95 per cent of normal galaxies.We find that stellar mass and kinematic asymmetry are inversely correlated and that kinematic asymmetry is both more frequent and stronger in low-mass galaxies. This builds on previous studies that found high fractions of kinematic asymmetry in low-mass galaxies using a variety of different methods. Concentration of star formation and kinematic disturbance are found to be correlated, confirming results found in previous work. This effect is stronger for high-mass galaxies (log(M*) > 10) and indicates that kinematic disturbance is linked to centrally concentrated star formation. Comparison of the inner (within 0.5Re) and outer Ha equivalent widths of asymmetric and normal galaxies shows a small but significant increase in inner equivalent width for asymmetric galaxies.