We consider the problem of counting lattice points contained in domains in $\mathbb{R}^d$ defined by products of linear forms and we show that the normalized discrepancies in these counting problems satisfy non-degenerate Central Limit Theorems, provided that $d \geq 9$. We also study more refined versions pertaining to "spiraling of approximations". Our techniques are dynamical in nature and exploit effective exponential mixing of all orders for actions of higher-rank abelian groups on the space of unimodular lattices.