We consider the magnetic Schrödinger operator with a variable metric in a two-dimensional simply connected periodic waveguide. All the coefficients are assumed to be periodic along the waveguide. We investigate the Dirichlet and Neumann boundary problems, as well as the boundary problem of the third type. Under wide conditions on the boundary of the waveguide providing a band structure of the spectrum, we prove the absolute continuity of the spectrum. Bibliography: 16 titles. © 2005 Springer Science+Business Media, Inc.