  We study the boundedness of Toeplitz operators with locally integrable
symbols on Bergman spaces $A^p(\Omega),$ $1<p<\infty,$ where $\Omega\subset
\mathbb{C}$ is a bounded simply connected domain with polygonal boundary. We
give sufficient conditions for the boundedness of generalized Toeplitz
operators in terms of "averages" of symbol over certain Cartesian squares. We
use the Whitney decomposition of $\Omega$ in the proof. We also give examples
of bounded Toeplitz operators on $A^p(\Omega)$ in the case where polygon
$\Omega$ has such a large corner that the Bergman projection is unbounded.
