Context. Winds of asymptotic giant branch stars are assumed to be driven by radiation pressure on dust. Previously, this process has been modeled with detailed time-dependent simulations of atmospheres and winds assuming spherically symmetric flows. In such models kinetic energy is injected by a variable inner boundary (“piston”) simulating the effects of stellar pulsation. However, the dynamical processes in these atmospheres – convection and pulsations – are actually three-dimensional. Aims. We present and analyze first 3D radiation hydrodynamics simulations of the convective interior and the atmosphere of a typical AGB star. In particular, we check whether the piston description in the 1D wind models is compatible with the 3D results. Methods. We used two different RHD codes, one (CO5BOLD) to produce 3D models of the outer convective envelope and the inner atmosphere of an AGB star, the other to describe the atmosphere and the wind acceleration region, including dust formation and non-grey radiative transfer, but assuming spherically symmetric flows. From the movements of stellar surface layers in the 3D models, we derived a description for the variable inner boundary in the 1D models. Results. The 3D models show large convection cells and pulsations that give rise to roughly spherically expanding shock waves in the atmosphere, levitating material into regions which are cool enough to allow for dust formation. The atmospheric velocity fields have amplitudes and time scales close to the values that are necessary to start dust formation in the 1D wind models. Conclusions. The convection cells in the 3D simulations are so large that the associated shock fronts appear almost spherical, justifying the assumption of spherical symmetry and the use of a piston boundary condition in the context of wind models. Nevertheless, certain non-radial structures exist in the dust shell developing in the 3D simulations which should be detectable with current interferometric techniques.