  We study the boundedness of Toeplitz operators $T_\psi$ with locally
integrable symbols on weighted harmonic Bergman spaces over the unit ball of
$\mathbb{R}^n$. Generalizing earlier results for analytic function spaces, we
derive a general sufficient condition for the boundedness of $T_\psi$ in terms
of suitable averages of its symbol. We also obtain a similar "vanishing"
condition for compactness. Finally, we show how these results can be
transferred to the setting of the standard weighted Bergman spaces of analytic
functions.
