  We obtain real-valued, time-periodic and radially symmetric solutions of the
cubic Klein-Gordon equation \begin{align}
  \partial_t^2 U - \Delta U + m^2 U = \Gamma (x) U^3
  \quad \text{on } \mathbb{R} \times \mathbb{R}^3, \end{align} which are weakly
localized in space. Various families of such "breather" solutions are shown to
bifurcate from any given nontrivial stationary solution. The construction of
weakly localized breathers in three space dimensions is, to the author's
knowledge, a new concept and based on the reformulation of the cubic
Klein-Gordon equation as a system of coupled nonlinear Helmholtz equations
involving suitable conditions on the far field behavior.
