A systematic study is carried out of a Lorentz lattice gas in order to model the growth dynamics of order-disorder interfaces. In the model, a particle, initially at the origin, moves on the bonds of an initially ordered square lattice, with sites covered by periodically repeated square blocks of 1, 4, or 9 right or left scattering rotators, whose orientations change after collisions with the particle. Depending then on the initial conditions of the blocks and the particle, one observes the following: (a) the particle randomizes the rotator orientations completely, in an ever growing disordered liquid phase inside the ordered solid phase on the rest of the lattice; (b) the particle propagates suddenly after a transient randomization period as in (a); or (c) the particle propagates through the ordered lattice immediately. A simple picture for the growth of the randomized region, which proceeds via an interface of fractal dimension 0.75, is discussed. The nature of the propagation for the cases mentioned can be modified by collisions with impurities.