We study analytically and numerically 'accessible' spatiotemporal solitons in a three-dimensional strongly nonlocal nonlinear medium. A general localized soliton solution of the 'acceptable' type is obtained in the Cartesian coordinates, using even and odd parabolic-cylinder functions. Characteristics of these accessible spatiotemporal solitons are discussed. The validity of the analytical solutions and their stability is verified by means of direct numerical simulations. 2013 IOP Publishing Ltd.