This paper presents a three-dimensional time-domain numerical model of the vibration and acoustic radiation from a guitar. The model involves the transverse displacement of the string excited by a force pulse, the flexural motion of the soundboard, and the sound radiation. A specific spectral method is used for solving the Kirchhoff-Love's dynamic top plate model for a damped, heterogeneous orthotropic material. The air-plate interaction is solved with a fictitious domain method, and a conservative scheme is used for the time discretization. Frequency analysis is performed on the simulated sound pressure and plate velocity waveforms in order to evaluate quantitatively the transfer of energy through the various components of the coupled system: from the string to the soundboard and from the soundboard to the air. The effects of some structural changes in soundboard thickness and cavity volume on the produced sounds are presented and discussed. Simulations of the same guitar in three different cases are also performed: "in vacuo," in air with a perfectly rigid top plate, and in air with an elastic top plate. This allows comparisons between structural, acoustic, and structural-acoustic modes of the instrument. Finally, attention is paid to the evolution with time of the spatial pressure field. This shows, in particular, the complex evolution of the directivity pattern in the near field of the instrument, especially during the attack.