The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions areinvestigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear strainhardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinctmaterial characteristic lengths. It can also capture the strong size effects arising at small scales, which results from theunderlying microstructures. The effects of microstructure on Mode III crack tip fields mainly consist in a substantial increase inthe singularities of the skew-symmetric stress and couple stress fields, which occurs also for small values of the strainhardening coefficient, whereas the symmetric stress field turns out to be non-singular according to the asymptotic crack tipfields for a stationary crack provided by the indeterminate theory of couple stress elasticity. The performed asymptotic analysisthus predicts a significant increase of the tractions level ahead of the crack-tip, due to the contribution of the rotation gradient.