Solar-type young stellar objects undergo periodic, energetic outbursts that appear to be the result of enhanced mass accretion driven by the gravitational instability of their disks. Such FU Orionis outbursts may have profound consequences for the earliest solids in a protoplanetary disk, namely the refractory inclusions containing abundant calcium and aluminum (CAIs). We present models of the orbital evolution of centimeter-radius particles representing large CAIs in marginally gravitationally unstable disks. The hydrodynamical evolution of the disks is calculated with a fully three-dimensional code, including compressional heating and cooling in the beta cooling approximation. The particles are initially distributed uniformly throughout the disk, which extends from 1 to 10 au around a solar-mass protostar, but within ∼100 yr the particles are concentrated by gas drag into regions surrounding the spiral arms and rings formed by the gas disk. The particles settle down toward the disk midplane, only to be lofted repeatedly upward by shock fronts. Large-scale radial transport both outward and inward occurs, with significant numbers of particles reaching the outer disk (∼10 au) and surviving for considerably longer times than would be the case in a quiescent disk with gas pressure monotonically decreasing with distance from the protostar. Individual particles experience wide ranges of disk temperatures during their journeys, ranging from 60 K in the outer disk to nearly 2000 K in spiral features. Future work will consider the implications for CAI rims of the thermochemical processing experienced during FU Orionis outbursts.