Discrete Diffusion Monte Carlo (DDMC) is a technique for increasing the efficiency of Monte Carlo simulations in diffusive media. If standard Monte Carlo is employed in such a regime, particle histories will consist of many small steps, a situation that results in a computationally inefficient calculation. In DDMC, particles take discrete steps between spatial cells according to a discretized diffusion equation. Each discrete step replaces many smaller Monte Carlo steps, thus increasing the efficiency of the simulation. In addition, because DDMC is based on the diffusion approximation, it should yield accurate solutions if used judiciously. In this paper, we present a new DDMC method for linear, steady-state radiation transport on adaptive-refinement meshes in two-dimensional Cartesian geometry. Adaptive-refinement meshes are characterized by local refinement such that a spatial cell may have multiple neighboring cells across each face. We specifically examine the cases of (a) a regular mesh structure without refinement, (b) a refined mesh structure where neighboring cells differ in refinement, and (c) a boundary mesh structure representing the interface between a diffusive region (where DDMC is used) and a nondiffusive region (where standard Monte Carlo is employed). With numerical examples, we demonstrate that our new DDMC technique is accurate and can provide efficiency gains of two orders of magnitude over standard Monte Carlo.