  We study Bergman kernels $K_\Pi$ and projections $P_\Pi$ in unbounded planar
domains $\Pi$, which are periodic in one dimension. In the case $\Pi$ is simply
connected we write the kernel $K_\Pi$ in terms of a Riemann mapping $\varphi$
related to the bounded periodic cell $\varpi$ of the domain $\Pi$. We also
introduce and adapt to the Bergman space setting the Floquet transform
technique, which is a standard tool for elliptic spectral problems in periodic
domains. We investigate the boundedness properties of the Floquet transform
operators in Bergman spaces and derive a general formula connecting $P_\Pi$ to
a projection on a bounded domain. We show how this theory can be used to
reproduce the above kernel formula for $K_\Pi$. Finally, we consider weighted
$L^p$-estimates for $P_\Pi$ in periodic domains.
