Biophysically detailed and anatomically realistic atrial models are emerging as a valuable tool in the study of atrial arrhythmias, nevertheless clinical use of these models would be favored by a reduction of computational times. This paper introduces a novel adaptive mesh algorithm, based on multiresolution representation (MR), for the efficient integration of cardiac ordinary differential equation (ODE)-partial differential equation (PDE) systems on unstructured triangle meshes. The algorithm applies a dynamically adapted node-centered finite volume method (FVM) scheme for integration of diffusion. The method accuracy and efficiency were evaluated by simulating propagation scenarios of increasing complexity levels (pacing, stable spirals, atrial fibrillation) on tomography-derived three-dimensional monolayer atrial models, based on a monodomain reaction-diffusion formulation coupled with the Courtemanche atrial ionic model. All simulated propagation patterns were accurately reproduced with substantially reduced computational times (10%-30% of the full-resolution simulation time). The proposed algorithm, combining the MR computational efficiency with the geometrical flexibility of unstructured meshes, may favor the development of patient-specific multiscale models of atrial arrhythmias and their application in the clinical setting.