  In the present paper we study Gelfand-Tsetlin modules defined in terms of BGG
differential operators. The structure of these modules is described with the
aid of the Postnikov-Stanley polynomials introduced in [PS09]. These
polynomials are used to identify the action of the Gelfand-Tsetlin subalgebra
on the BGG operators. We also provide explicit bases of the corresponding
Gelfand-Tsetlin modules and prove a simplicity criterion for these modules. The
results hold for modules defined over standard Galois orders of type $A$ - a
large class of rings that include the universal enveloping algebra of
$\mathfrak{gl} (n)$ and the finite $W$-algebras of type $A$.
