Let $k=(k_\alpha)_{\alpha\in \cal R}$ be a positive-real valued multiplicity function related to a root system $\cal R$, and $\Delta_k$ be the Dunkl-Laplacian operator. For $(x,t)\in \mathbb{R}^N\times \mathbb{R}$, denote by $u_k(x,t)$ the solution to the deformed wave equation $\Delta_k^xu_k(x,t)=\partial_{tt}u_k(x,t)$, where the initial data belong to the Schwartz space on $\mathbb{R}^N$. We prove that for $k\geq 0$ and $N\geq 1$, the wave equation satisfies a weak Huygens' principle, and only if $(N-3)/2+\sum_{\alpha\in \cal R^+}k_\alpha\in \mathbb{N}$, a strict Huygens' principle holds. Here $\cal R^+\subset \cal R $ is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius $R>0$ about the origin, the strict Huygens' principle implies that the support of $u_k(x,t)$ is contained in the conical shell $\lbrace(x,t)\in \mathbb{R}^N\times \mathbb{R}\;|\; | t| -R \leq \left\| x \right\| \leq | t |+R\rbrace$. Our approach uses the representation theory of the group $SL(2,\R)$, and Paley-Wiener theory for the Dunkl transform. Also, we show that the ($t$-independent) energy functional of $u_k$ is, for large $| t|$, partitioned into equal potential and kinetic parts.