The photodetachment of negative ions inside a two-dimensional microcavity has been studied by many researchers. As to the photodetachment of negative ions in the three-dimensional microcavity, the research is relatively little. In this paper, we study the photodetachment cross-section of $\rm{H}^{−}$ ion inside a three-dimensional cubical microcavity for the first time.We have observed the classical dynamics of the photodetached electron inside the cubical microcavity and found out its closed orbits. Then we calculate the photodetachment cross-section of this system. It is shown that owing to the interference effects of the electron wave travelling along various closed orbits, oscillatory structures appear in the photodetachment cross-section. And the oscillatory structures depend on the laser polarization sensitively. Compared to the photodetachment of $\rm{H}^{−}$ ion inside a square microcavity, in photodetachment of $\rm{H}^{−}$ ion in cubical cavity the number of the closed orbits is increased and the oscillatory structure in the photodetachment cross-section becomes much more complex. Through our study, researchers can gain a deep understanding on the correspondence of the classical dynamics and the quantum mechanics. Our study may guide future experimental research in the field of the photodetachment electron dynamics inside a three-dimensional microcavity.