Let G be a classical p-adic group. We are interested in the K-theory of the reduced C*-algebra C*_r (G). Via the Baum--Connes conjecture, these K-groups can in principle be computed as equivariant K-homology of the Bruhat--Tits building of G. But in practice this is hopeless. Instead, we approach the problem via the representation theory of G, which quickly leads us to affine Hecke algebras. We will discuss the C*-completions of affine Hecke algebras, and we explain why their K-theory does not depend on the deformation parameters q. An affine Hecke algebra with q=1 is just the crossed product for a Weyl group W acting on a torus T. We will compute its topological K-theory (including torsion) in terms of T and W. This enables us to express the K-theory of C*_r (G) in terms of the tori and finite groups associated to Bernstein components.