We explore families of spatiotemporal dissipative solitons in a model of three-dimensional (3D) laser cavities including a combination of gain, saturable absorption, and transverse grating. The model is based on the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity and a two-dimensional (2D) periodic potential representing the grating. Fundamental and vortical solitons are found in a numerical form as attractors in this model and their stability against strong random perturbations is tested by direct simulations. The fundamental solitons are completely stable while the vortices, built as rhombus-shaped complexes of four fundamental solitons, may be split by perturbations into their constituents separating in the temporal direction. Nevertheless, a sufficiently strong grating makes the vortices practically stable objects.