The authors study the Jordan normal forms of the local and global monodromies over complete intersection subvarieties of \({\mathbb C}^n\) by using the theory of motivic Milnor fibers and their Hodge realizations developed by J. Denef and F. Loeser [J. Algebr. Geom. 7, No. 3, 505–537 (1998; Zbl 0943.14010); Prog. Math. 201, 327–348 (2001; Zbl 1079.14003)], G. Guibert, F. Loeser and M. Merle [Duke Math. J. 132, No. 3, 409–457 (2006; Zbl 1173.14301)], Y. Matsui and the second author [“Monodromy at infinity of polynomial maps and Newton polyhedra (with appendix by C. Sabbah)”, Preprint, arXiv:0912.5144]. The results are explicitly described by the mixed volumes of the faces of Newton polyhedra.