  We study the $T$-violating lepton transverse polarization ($P^\perp_l$) in
three body semileptonic heavy meson decays to pseudoscalar mesons and to vector
mesons. We calculate these polarizations in the heavy quark effective limit,
which simplifies the expressions considerably. After examining constraints from
$CP$ conserving (including $b \rightarrow s \gamma$) and $CP$ violating
processes, we find that in $B$ decays, $P^\perp$ of the muon in multi-Higgs
doublet models can be of order $10\%$, while $P^\perp$ of the $\tau$ can even
approach unity. In contrast, $P^\perp_\mu$ in $D$ decays is at most 1.5\%. We
discuss possibilities for detection of $P^\perp_l$ at current and future $B$
factories. We also show that $P^\perp_l$ in decays to vector mesons, unlike in
decays to pseudoscalars, can get contributions from left-right models.
Unfortunately, $P^\perp_l$ in that case is proportional to $W_L$-$W_R$ mixing,
and is thus small.
