This paper presents a methodology to compute acoustic damping rates of transversal, high-frequency modes induced by vortex-shedding. The acoustic damping rate presents one key quantity for the assessment of the linear thermoacoustic stability of gas turbine combustors. State-of-the-art network models—as employed to calculate damping rates in low-frequency, longitudinal systems—cannot fulfill this task due to the acoustic noncompactness encountered in the high-frequency regime. Furthermore, it is yet unclear, whether direct eigensolutions of the linearized Euler equations (LEE), which capture the mechanism of vortex shedding, yield correct damping rate results constituted by the implicit presence of acoustic as well as hydrodynamic contributions in these solutions. The methodology's applicability to technically relevant systems is demonstrated by a validation test case using a lab-scale, swirl-stabilized combustion system.