Trapped modes in the linearized water-wave problem are free oscillations of finite energy in an unbounded
fluid with a free surface. It has been known for some time that such modes are supported by certain
structures when held fixed, but recently it has been demonstrated that in two dimensions trapped modes
are also possible for freely-floating structures that are able to respond to the hydrodynamic forces acting
upon them. For a freely-floating structure such a mode is a coupled oscillation of the fluid and the structure
that, in the absence of viscosity, persists for all time. Here previous work on the two-dimensional problem
is extended to give motion trapping structures in the three-dimensional water-wave problem that have a
vertical axis of symmetry.