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We describe a framework for numerical calculation of time-periodically oscillating (breather) solutions to the discretized Lugiato-Lefever equation (LLE), as well as their linear stability as obtained from numerical Floquet analysis. Compared to earlier approaches, our work allows for the following conclusions: (i) The complete families of solutions are obtained also in regimes of instability; (ii) analysis of Floquet spectra and the corresponding eigenvectors show clearly the nature of the various Hopf and period-doubling bifurcations; (iii) properties of breather solutions to the continuous LLE are connected to corresponding oscillating solutions of the discrete LLE, which is of interest in its own right modeling coupled nonlinear cavities. In particular, we show that the oscillating discrete cavity solitons found in earlier work can be viewed as lattice versions of the continuous LLE breathers, as there is a smooth continuation in parameter space connecting them. Moreover, we confirm the existence of stable breathers at large detunings that was recently observed experimentally and describe their appearance from bifurcations.