  We determine the periodic cyclic homology of the Iwahori-Hecke algebras
$\Hecke_q$, for $q \in \CC^*$ not a ``proper root of unity.'' (In this paper,
by a {\em proper root of unity} we shall mean a root of unity other than 1.)
Our method is based on a general result on periodic cyclic homology, which
states that a ``weakly spectrum preserving'' morphism of finite type algebras
induces an isomorphism in periodic cyclic homology. The concept of a weakly
spectrum preserving morphism is defined in this paper, and most of our work is
devoted to understanding this class of morphisms. Results of Kazhdan--Lusztig
and Lusztig show that, for the indicated values of $q$, there exists a weakly
spectrum preserving morphism $\phi_q : \Hecke_q \to J$, to a fixed finite type
algebra $J$. This proves that $\phi_q$ induces an isomorphism in periodic
cyclic homology and, in particular, that all algebras $\Hecke_q$ have the same
periodic cyclic homology, for the indicated values of $q$. The periodic cyclic
homology groups of the algebra $\Hecke_1$ can then be determined directly,
using results of Karoubi and Burghelea, because it is the group algebra of an
extended affine Weyl group.
