There has been some recent speculation that a connection may exist between the quasinormal-mode spectra of highly damped black holes and the fundamental theory of quantum gravity. This notion follows from a conjecture by Hod that the real part of the highly damped mode frequencies can be used to calibrate the semi-classical level spacing in the black hole quantum area spectrum. However, even if the level spacing can be fixed in this manner, it still remains unclear whether this implies a physically significant "duality" or merely a numerical coincidence. This tapestry of ideas serves as the motivation for the current paper. We utilize the "monodromy approach" to calculate the quasinormal-mode spectra for a generic class of black holes in two-dimensional dilatonic gravity. Our results agree with the prior literature whenever a direct comparison is possible and provide the analysis of a much more diverse class of black hole models than previously considered.