  We give a detailed and self-contained description of a recently developed
theoretical and numerical method for the simulation of three identical bosonic
alkali-metal atoms near a Feshbach resonance, where the Efimov effect is
induced. The method is based on a direct construction of the off-shell two-body
transition matrix from exact eigenfunctions of the embedded two-body
Hamiltonians, obtained using realistic parameterizations of the interaction
potentials which accurately reproduce the molecular energy levels. The
transition matrix is then inserted into the appropriate three-body integral
equations, which may be efficiently solved on a computer. We focus especially
on the power of our method in including rigorously the effects of multichannel
physics on the three-body problem, which are usually accounted for only by
various approximations. We demonstrate the method for ${}^7$Li, where we
recently showed that a correct inclusion of this multichannel physics resolves
the long-standing disagreement between theory and experiment regarding the
Efimovian three-body parameter. We analyze the Efimovian enhancement of the
three-body recombination rate on both sides of the Feshbach resonance,
revealing strong sensitivity to the spin structure of the model thus indicating
the prevalence of three-body spin-exchange physics. Finally, we discuss an
extension of our methodology to the calculation of three-body bound-state
energies.
