The paper is devoted to the relationship between the spectra of the 1D Dirac operator with periodic potential and the smoothness of the potential. The relationship between the rate of decay of sequences built up from the spectral points of the operator and the smoothness of the potential is analyzed. Periodic, antiperiodic and Dirichlet boundary conditions are considered. The smoothness of the potential is measured by weight sequences of several different types. The results of the paper are valid for any \(L^2\)-potentials, not only for selfadjoint ones. Similar results for the Schrödinger operators were obtained earlier by the authors in [J. Funct. Anal. 195, 89–128 (2002; Zbl 1037.34080)].