A new approach to calculating Floquet spectra of multilayered periodic waveguides is presented. The problem is formulated as an eigenvalue problem of the Helmholtz equation on an infinite strip with discontinuous wavenumber. The strip is decomposed into a rectangle and two semi-infinite domains, and the problem is reduced to a nonlinear eigenvalue problem involving Dirichlet-to-Neumann (DtN) operators on the interfaces of the domains. A solution scheme based on the Taylor expansion of the DtN operator with respect to the Floquet exponent, whose order of convergence can be made arbitrarily large, is derived. An application to a typical waveguide geometry demonstrates the efficiency and accuracy of the approach.