The notion of Bernstein-- Sato polynomial has been extended to the case of arbitrary subvarieties in an smooth affine variety by Budur, Mustata, and Saito. In this talk, I will explain how one can further extend their definition to the case of arbitrary ideals on toric varieties by using the combinatorial description (obtained by Musson and Jones independently) of Grothendieck's ring of differential operators on toric varieties. In particular, I will show that the relation between the roots of Bernstein-- Sato polynomials and the jumping coefficients of multiplier ideals can also be extended to the toric setting (at least in the case of monomial ideals). This is based on a joint work in progress with Laura Matusevich.