In this paper, we investigate an electrodynamics in which the physical modes are coupled to a Lorentz-violating background by means of a higher-derivative term. We analyze the modes associated with the dispersion relations obtained from the poles of the propagator. More specifically, we study Maxwell's electrodynamics modified by a Lorentz-violating operator of mass dimension 6. The modification has the form Dβα∂σFσβ∂λFλα; i.e., it possesses two additional derivatives coupled to a CPT-even tensor Dβα that plays the role of the fixed background. We first evaluate the propagator and obtain the dispersion relations of the theory. By doing so, we analyze some configurations of the fixed background and search for sectors where the energy is well defined and causality is assured. A brief analysis of unitarity is included for particular configurations. Afterward, we perform the same kind of analysis for a more general dimension-6 model. We conclude that the modes of both Lagrange densities are possibly plagued by physical problems, including causality and unitarity violation, and that signal propagation may become physically meaningful only in the high-momentum regime.