Numerical studies of the propagation of ultrashort optical pulses in quadratic nonlinear media are presented. Two- parameter families of chirped stationary spatiotemporal solitons in dispersive quadratically nonlinear media are constructed in the presence of temporal walkoff. It is found that the walking spatiotemporal solitons are dynamically stable in most cases. We find also one-parameter families of spinning spatiotemporal bright solitons in dispersive media with quadratic nonlinearities and study their stability. The comparison with a simple variational approximation for the spinning spatiotemporal solitons demonstrates that, though the variational approximation is not very accurate, it correctly describes the qualitative features of the spinning spatiotemporal solitons. We show that the spinning spatiotemporal solitons are subject to a storing azimuthal instability. The instability breaks the spinning soliton into stable nonspinning light bullets.