In this article we study Hardy spaces ${\mathcal{H}}^{p,q}\left({\mathbb{R}}^d\right)$, $0<p,q<\infty$, modeled over amalgam spaces $(L^p,{\ell }^q)\left({\mathbb{R}}^d\right)$. We characterize ${\mathcal{H}}^{p,q}\left({\mathbb{R}}^d\right)$ by using first-order classical Riesz transforms and compositions of first-order Riesz transforms, depending on the values of the exponents $p$ and $q$. Also, we describe the distributions in ${\mathcal{H}}^{p,q}\left({\mathbb{R}}^d\right)$ as the boundary values of solutions of harmonic and caloric Cauchy–Riemann systems. We remark that caloric Cauchy–Riemann systems involve fractional derivatives in the time variable. Finally, we characterize the functions in $L^2\left({\mathbb{R}}^d\right)\cap {\mathcal{H}}^{p,q}\left({\mathbb{R}}^d\right)$ by means of Fourier multipliers $m_{\theta }$ with symbol $\theta (\cdot /|\cdot \left|\right)$, where $\theta \in C^{\infty }\left({\mathbb{S}}^{d-1}\right)$ and ${\mathbb{S}}^{d-1}$ denotes the unit sphere in ${\mathbb{R}}^d$.