  In this work we extend the theory of the classical Hardy space $H^1$ to the
rational Dunkl setting. Specifically, let $\Delta$ be the Dunkl Laplacian on a
Euclidean space $\mathbb{R}^N$. On the half-space
$\mathbb{R}_+\times\mathbb{R}^N$, we consider systems of conjugate
$(\partial_t^2+\Delta_{\mathbf{x}})$-harmonic functions satisfying an
appropriate uniform $L^1$ condition. We prove that the boundary values of such
harmonic functions, which constitute the real Hardy space $H^1$, can be
characterized in several different ways, namely by means of atoms, Riesz
transforms, maximal functions or Littlewood-Paley square functions.
