In contexts such as suspension feeding in marine ecologies there is an interplay between Brownian motion of nonmotile particles and their advection by flows from swimming microorganisms. As a laboratory realization, we study passive tracers in suspensions of eukaryotic swimmers, the alga Chlamydomonas reinhardtii. While the cells behave ballistically over short intervals, the tracers behave diffusively, with a time-dependent but self-similar probability distribution function of displacements consisting of a Gaussian core and robust exponential tails. We emphasize the role of flagellar beating in creating oscillatory flows that exceed Brownian motion far from each swimmer.