  We present a thorough experimental study of the three-dimensional
hyperhoneycomb Kitaev magnet $\beta$-Li$_2$IrO$_3$, using a combination of
inelastic neutron scattering (INS), time-domain THz spectroscopy, and heat
capacity measurements. The main results include a massive low-temperature
reorganization of the INS spectral weight that evolves into a broad peak
centered around 12 meV, and a distinctive peak in the THz data at 2.8(1) meV. A
detailed comparison to powder-averaged spin-wave theory calculations reveals
that the positions of these two features are controlled by the anisotropic
$\Gamma$ coupling and the Heisenberg exchange $J$, respectively. The refined
microscopic spin model places $\beta$-Li$_2$IrO$_3$ in close proximity to the
Kitaev spin liquid phase.
