The movement of motor particles consisting of one or several molecular motors bound to a cargo particle is studied theoretically. The particles move on patterns of immobilized filaments. Several patterns are described for which the motor particles undergo nondirected but enhanced diffusion. Depending on the walking distance of the particles and the mesh size of the patterns, the active diffusion coefficient exhibits three different regimes. For micrometer-sized motor particles in water, e.g., this diffusion coefficient can be enhanced by 2 orders of magnitude.