This is the first part of a two-part series dealing with complex dipolar waves propagating along the axes of 1D, 2D, and 3D infinite periodic arrays of small lossless and lossy permeable spheres. The theory is presented in this paper and numerical results are presented by Shore and Yaghjian (2012). The focus is on the dispersion (k–β) equations relating the array propagation constant, β, to the free-space wave number,k, for dipolar complex waves. The k–β equation for the complex propagation constants of a given array is obtained from the corresponding equation previously obtained for the real propagation constants by rewriting the real propagation dispersion equation in a form that can be analytically continued into the complex β plane. This equation reduces correctly to the real β dispersion equation and enables complex values of β to be found as a function of the array element parameters. By allowing for all the possible branches of the multivalued homogeneous dispersion equation analytically continued into the complex βplane, the propagation constants of all the improper as well as proper complex waves supported by the 1D, 2D, and 3D arrays are found from the homogeneous solutions for these arrays. Green's functions for external sources are not required to find the propagation constants of the complex waves supported by the arrays. For 3D arrays, in certain frequency ranges, it is possible to regard the arrays as media characterized by bulk or effective permittivities and permeabilities. Expressions for these bulk parameters, more accurate than the Clausius-Mossotti expressions, are obtained from quantities readily available in the solutions of the dispersion equations.