The aim of this paper is to investigate the enhancement of the damping ratio of a structure with embedded microbeam resonators in air-filled internal cavities. In this context, we discuss theoretical aspects in the framework of the effective modal damping ratio (MDR) and derive an approximate relation expressing how an increased damping due to the acoustic medium surrounding the microbeam affect the MDR of the macrobeam. We further analyze the effect of including dissipation of the acoustic medium by using finite element (FE) analysis with acoustic-structure interaction (ASI) using a simple phenomenological acoustic loss model. An eigenvalue analysis is carried out to demonstrate the improvement of the damping characteristic of the macrobeam with the resonating microbeam in the lossy air and the results are compared to a forced vibration analysis for a macrobeam with one or multiple embedded microbeams. Finally we demonstrate the effect of randomness in terms of position and size of microbeams and discuss the difference between the phenomenological acoustic loss model and a full thermoacoustic model.