The diagonal forms are constructed for the translation operators for the Helmholtz equation in three dimensions. While the operators themselves have a fairly complicated structure (described somewhat incompletely by the classical addition theorems for the Bessel functions), their diagonal forms turn out to be quite simple. These diagonal forms are realized as generalized integrals, possess straightforward physical interpretations, and admit stable numerical implementation. This paper uses the obtained analytical apparatus to construct an algorithm for the rapid application to arbitrary vectors of matrices resulting from the discretization of integral equations of the potential theory for the Helmholtz equation in three dimensions. It is an extension to the three-dimensional case of the results of Rokhlin (J. Complexity4(1988), 12-32), where a similar apparatus is developed in the two-dimensional case.