We study motivic zeta functions of degenerating families of Calabi-Yau
varieties. Our main result says that they satisfy an analog of Igusa's
monodromy conjecture if the family has a so-called Galois-equivariant Kulikov
model; we provide several classes of examples where this condition is verified.
We also establish a close relation between the zeta function and the skeleton
that appeared in Kontsevich and Soibelman's non-archimedean interpretation of
the SYZ conjecture in mirror symmetry.