The answer is nice for flat tori in every dimension. Let's write such a torus as $V/\Gamma$ where $V$ is a finite-dimensional real vector space of dimension $n$ with inner product $\langle \cdot, \cdot \rangle$ and $\Gamma$ is a lattice (a discrete subgroup isomorphic to $\mathbb{Z}^n$ which spans $V$). Any twice-differentiable eigenfunction $f : V/\Gamma \to \mathbb{C}$ of the Laplacian is in particular a bounded eigenfunction of the Laplacian on $V$, so we can take it to have the form $f_w(v) = e^{2 \pi i \langle w, v \rangle}$
for some $w \in V$ (for reasons to be described later). We also need to impose the constraint that $f_w$ is invariant under $\Gamma$, hence that $e^{2\pi i \langle w, v \rangle} = e^{2 \pi i \langle w, v + g \rangle}$
for every $g \in \Gamma$. This condition is satisfied if and only if $w$ belongs to the dual lattice $\Gamma^{\vee}$, which consists of all vectors $w$ such that $\langle w, g \rangle \in \mathbb{Z}$ for all $g \in \Gamma$. Moreover, $\Delta f_w = - 4 \pi^2 \| w \|^2$
so the eigenvalues of the Laplacian on $V/\Gamma$ are just $- 4\pi^2$ times the squares of the lengths of the vectors in $\Gamma^{\vee}$.