  Dirac materials respond to lattice deformations as if the electrons were
coupled to gauge fields. We derive the elastic gauge fields in the
hyperhoneycomb lattice, a three dimensional (3D) structure with trigonally
connected sites. In its semimetallic form, this lattice is a nodal-line
semimetal with a closed loop of Dirac nodes. Using strain engineering, we find
a whole family of strain deformations that create uniform nearly flat Landau
levels in 3D. We propose that those Landau levels can be created and tuned in
metamaterials with the application of a simple uniaxial temperature gradient.
In the 3D quantum anomalous Hall phase, which is topological, we show that the
components of the elastic Hall viscosity tensor are multiples of
$\eta_{H}=\beta^{2}\sqrt{3}/\left(8\pi a^{3}\right)$, where $\beta$ is an
elastic parameter and $a$ is the lattice constant.
