  We consider $G_2$-manifolds with an effective torus action that is
multi-Hamiltonian for one or more of the defining forms. The case of
$T^3$-actions is found to be distinguished. For such actions multi-Hamiltonian
with respect to both the three- and four-form, we derive a Gibbons-Hawking type
ansatz giving the geometry on an open dense set in terms a symmetric $3\times
3$-matrix of functions. This leads to particularly simple examples of explicit
metrics with holonomy equal to $G_2$. We prove that the multi-moment maps
exhibit the full orbit space topologically as a smooth four-manifold containing
a trivalent graph as the image of the set of special orbits and describe these
graphs in some complete examples.
