  We consider tilings $(\mathcal{Q},\Phi)$ of $\mathbb{R}^d$ where
$\mathcal{Q}$ is the $d$-dimensional unit cube and the set of translations
$\Phi$ is constrained to lie in a pre-determined lattice $A \mathbb{Z}^d$ in
$\mathbb{R}^d$. We provide a full characterization of matrices $A$ for which
such cube tilings exist when $\Phi$ is a sublattice of $A\mathbb{Z}^d$ with any
$d \in \mathbb{N}$ or a generic subset of $A\mathbb{Z}^d$ with $d\leq 7$. As a
direct consequence of our results, we obtain a criterion for the existence of
linearly constrained frequency sets, that is, $\Phi \subseteq A\mathbb{Z}^d$,
such that the respective set of complex exponential functions $\mathcal{E}
(\Phi)$ is an orthogonal Fourier basis for the space of square integrable
functions supported on a parallelepiped $B\mathcal{Q}$, where $A, B \in
\mathbb{R}^{d \times d}$ are nonsingular matrices given a priori. Similarly
constructed Riesz bases are considered in a companion paper.
