By implementing a version of the dissipative mapping technique introduced by R. Malhotra, we have developed a new integration code for the N-body problem that incorporates the effects of radiation pressure, Poynting-Robertson (P-R) drag, and solar wind drag. The advantage of employing the dissipative mapping technique is that it modifies the basic N-body symplectic integration algorithm developed by Wisdom & Holman to allow certain nongravitational effects to be modeled and therefore retains the speed of execution common to codes based upon this algorithm. To achieve this, we have adapted the dissipative mapping technique to the requirements of the forces being modeled. We present the results of tests that demonstrate the suitability of this new dissipative integration code for investigating the dynamical behavior of micron-sized dust particles in heliocentric orbits in the solar system and, more generally, of particles in exosolar planetary systems where the dominant nongravitational perturbations to the particles' astrocentric orbits are due to the effects of radiation pressure, P-R drag, and solar wind drag.