In this paper we study the entire solutions of a class of
periodic Allen--Cahn equations
\begin{equation}
-\Delta u(x,y)+a(x)W^{'}(u(x,y)) = 0{,}\quad
(x,y) \in \mathbb{R}^{2},
\end{equation}
where $a(x)\colon \mathbb{R} \to \mathbb{R}^{+}$ is a periodic,
positive function and $W \in C^{2}(\mathbb{R}, \mathbb{R})$
is a double-well potential. We look for the entire
solutions of the above equation with asymptotic conditions
$u(x,y) \to \sigma_{\pm}$ as $x \to \pm\infty$ uniformly with
respect to $y \in \mathbb{R}$. Via variational methods we
find infinitely many solutions.