We derive a cluster mean-field theory for an Ising Hamiltonian using a cluster-lattice Fourier transform with a cluster of size Nc and a coarse-grained (CG) lattice into cells of size Ncell. We explore forms with Ncell⩾Nc, including a non-CG (NCG) version with Ncell→∞. For Nc=Ncell, the set of static, self-consistent equations relating cluster and CG lattice correlations is analogous to that in dynamical cluster approximation and cellular dynamical mean-field theory used in correlated electron physics. A variational Nc-site cluster grand potential based on Nc=Ncell CG lattice maintains thermodynamic consistency and improves predictions, recovering Monte Carlo and series expansion results upon finite-size scaling; notably, the Nc=1 CG results already predict well the first- and second-order phase boundary topology and transition temperatures for frustrated lattices. The NCG version is significantly faster computationally than the CG case and more accurate at fixed Nc for ferromagnetism, which is potentially useful for cluster expansion and quantum cluster applications.