We propose a novel approach to simulate the evolution of polycrystalline microstructures based upon a disconnection model for grain boundary (GB) kinetics. Disconnections are line defects that lie solely with GB and are characterized by both a Burgers vector and a step height, as set by the GB bicrystallography. The model incorporates surface tension, applied stress, and jumps in chemical potential across GBs. The model also includes disconnection nucleation and mobility. We first derive a continuum equation of motion for individual GBs, and then for GB triple junctions (TJ) that rigorously accounts for conservation of disconnection Burgers vectors and step heights and couples the GBs meeting at the TJ. We then implement this model in a continuum simulation of GB dynamics without TJs, with TJs and in a polycrystalline microstructure. The resultant simulations provide clear demonstrations of the importance of including a crystallography-respecting microscopic model of microstructure evolution and the intrinsic coupling between stress, capillarity, and microstructure connectivity in microstructure evolution.