In a computer simulation, dust grains of radius 10, 30, and 100 μm were released at perihelion passage from each of 35 different celestial bodies: 15 main belt asteroids, 15 short period comets with perihelion greater than 1 AU, and 5 short period comets with perihelion less than 1 AU. The evolving orbit of each of the 105 released dust grains was then continuosly computed with the Everhart numerical integrator until the orbit aphelion passed inside of 0.387 AU, or the dust grain had been ejected from the Solar System. The forces due to the gravity of the Sun and the planets as well as radiation pressure, Poynting-Robertson drag, and solar wind drag were all included in these numerical simulations. It is found that when dust grains evolve to intersection with the Earth's orbit, they nearly always retain orbital characteristics indicative of their origins; grains from main belt asteroids differ significantly in orbital characteristics, especially in orbital eccentricity, from grains that evolve from comets. Average intersect velocities with the Earth, before the Earth's gravitationa acceleration is taken into account, are about 5 km/sec for asteroidal and in excess of 12 km/ sec for cometary-derived grains. Average orbital eccentricities are about 0.1 for asteroidal grains and usually in excess of 0.4 for cometary grains when their orbits can intersect the Earth's orbit. These results mean that accurate trajectory measurements of meteoroids collected with a near-Earth space platform would make it possible to distinguish asteroidal grains from cometary grains.