We characterise regions in the complex plane that contain all non-embedded eigenvalues of a perturbed periodic Dirac operator on the real line with real-valued periodic potential and a generally non-symmetric matrix-valued perturbation V . We show that the eigenvalues are located close to the end-points of the spectral bands for small V\in L^{1}(\mathbb{R})^{2\times 2} , but only close to the spectral bands as a whole for small V\in L^{p}(\mathbb{R})^{2\times 2} , p > 1 . As auxiliary results, we prove the relative compactness of matrix multiplication operators in L^{2p}(\mathbb{R})^{2\times 2} with respect to the periodic operator under minimal hypotheses, and find the asymptotic solution of the Dirac equation on a finite interval for spectral parameters with large imaginary part.