  Iwahori-Hecke algebras for the infinite series of complex reflection groups
$G(r,p,n)$ were constructed recently in the work of Ariki and Koike, Brou\'e
and Malle, and Ariki. In this paper we give Murnaghan-Nakayama type formulas
for computing the irreducible characters of these algebras. Our method is a
generalization of that in our earlier paper in which we derived
Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of
the classical Weyl groups. In both papers we have been motivated by C. Greene,
who gave a new derivation of the Murnaghan-Nakayama formula for irreducible
symmetric group characters by summing diagonal matrix entries in Young's
seminormal representations. We use the analogous representations of the
Iwahori-Hecke algebra of $G(r,p,n)$ given by Ariki and Koike.
