The propagation of linear, dissipative, time‐harmonic acoustic waves in a broad class of periodic waveguides is investigated theoretically. It is shown that a Floquet‐like theorem is applicable to the system of partial differential equations and boundary conditions that describes the dynamics of a thermoviscous fluid in a periodic waveguide, and that the solution wave functions are therefore Bloch wave functions. Expressions for the parameters that characterize the Bloch wave functions are derived and the band structure of these functions of frequency is investigated. It is found that conditions of axial inversion symmetry (the waveguide may be isotropic or anisotropic) and reciprocity impose substantial restrictions on the allowed Bloch wave solutions.