A novel construction of lattices is proposed. This construction can be thought of as a special class of Construction A from codes over finite rings that can be represented as the Cartesian product of $L$ linear codes over $\mathbb{F}_{p_1},\ldots,\mathbb{F}_{p_L}$, respectively, and hence is referred to as Construction $π_A$. The existence of a sequence of such lattices that is good for channel coding (i.e., Poltyrev-limit achieving) under multistage decoding is shown. A new family of multilevel nested lattice codes based on Construction $π_A$ lattices is proposed and its achievable rate for the additive white Gaussian channel is analyzed. A generalization named Construction $π_D$ is also investigated which subsumes Construction A with codes over prime fields, Construction D, and Construction $π_A$ as special cases.