We propose a scheme for stabilizing spatiotemporal solitons (STSs) in media with cubic self-focusing nonlinearity and "dispersion management," i.e., a layered structure inducing periodically alternating normal and anomalous group-velocity dispersion. We develop a variational approximation for the STS, and verify results by direct simulations. A stability region for the two-dimensional (2D) STS (corresponding to a planar waveguide) is identified. At the borders between this region and that of decay of the solitons, a more sophisticated stable object, in the form of a periodically oscillating bound state of two subpulses, is also found. In the 3D case (bulk medium), all the spatiotemporal pulses spread out or collapse.