  Let $\mathbb{H}$ be the first Heisenberg group, and let $k \in
C^{\infty}(\mathbb{H} \, \setminus \, \{0\})$ be a kernel which is either odd
or horizontally odd, and satisfies $$|\nabla_{\mathbb{H}}^{n}k(p)| \leq
C_{n}\|p\|^{-1 - n}, \qquad p \in \mathbb{H} \, \setminus \, \{0\}, \, n \geq
0.$$ The simplest examples include certain Riesz-type kernels first considered
by Chousionis and Mattila, and the horizontally odd kernel $k(p) =
\nabla_{\mathbb{H}} \log \|p\|$. We prove that convolution with $k$, as above,
yields an $L^{2}$-bounded operator on regular curves in $\mathbb{H}$. This
extends a theorem of G. David to the Heisenberg group.
  As a corollary of our main result, we infer that all $3$-dimensional
horizontally odd kernels yield $L^{2}$ bounded operators on Lipschitz flags in
$\mathbb{H}$. This was known earlier for only one specific operator, the
$3$-dimensional Riesz transform. Finally, our technique yields new results on
certain non-negative kernels, introduced by Chousionis and Li.
