If the metric of an almost Kenmotsu manifold with conformal Reeb foliation is a gradient Ricci soliton, then it is an Einstein metric and the Ricci soliton is expanding. Moreover, let $(M^{2n+1},\phi,\xi,\eta,g)$ be an almost Kenmotsu manifold with $\xi$ belonging to the $(k,\mu)'$-nullity distribution and $h\neq0$. If the metric $g$ of $M^{2n+1}$ is a gradient Ricci soliton, then $M^{2n+1}$ is locally isometric to the Riemannian product of an $(n+1)$-dimensional manifold of constant sectional curvature $-4$ and a flat $n$-dimensional manifold, also, the Ricci soliton is expanding with $\lambda=4n$.