  Motivated by the recent synthesis of two insulating Li$_2$IrO$_3$ polymorphs,
where Ir$^{4+}$ $S_{eff}$=1/2 moments form 3D ("harmonic") honeycomb structures
with threefold coordination, we study magnetic Hamiltonians on the resulting
$\beta$-Li$_2$IrO$_3$ hyperhoneycomb lattice and $\gamma$-Li$_2$IrO$_3$
stripyhoneycomb lattice. Experimentally measured magnetic susceptibilities
suggest that Kitaev interactions, predicted for the ideal 90$^\circ$ Ir-O-Ir
bonds, are sizable in these materials. We first consider pure Kitaev
interactions, which lead to an exactly soluble 3D quantum spin liquid (QSL)
with emergent Majorana fermions and Z$_2$ flux loops. Unlike 2D QSLs, the 3D
QSL is stable to finite temperature, with $T_c \approx |K|/100$. On including
Heisenberg couplings, exact solubility is lost. However, by noting that the
shortest closed loop $\ell$ is relatively large in these structures, we
construct an $\ell\rightarrow \infty$ approximation by defining the model on
the Bethe lattice. The phase diagram of the Kitaev-Heisenberg model on this
lattice is obtained directly in the thermodynamic limit, using tensor network
states and the infinite-system time-evolving-block-decimation (iTEBD)
algorithm. Both magnetically ordered and gapped QSL phases are found, the
latter being identified by an entanglement fingerprint.
