The aim of this study is to develop a machine learning network to estimate the fundamental vibration period values of existing reinforced concrete (RC) buildings with damaged structural and non-structural elements. By considering the proposed machine learning network, changes in the fundamental vibration period of RC buildings due to potential damage states on structural members and infill walls are estimated. In this context, first of all, the level of reduction in stiffness caused by different damage levels in different types of structural elements is determined. Afterwards, an extensive database composed of 16,000 different building simulations with varying geometrical and mechanical properties is generated. 3D numerical models of these simulations are formed, and the fundamental vibration period values of the generated numerical models are determined. For each numerical model, a variant model at a certain damage state is also created by assigning predefined damage parameters to both structural and non-structural components. To this end, damage factor coefficients are used in stiffness matrices. An artificial neural network model is developed, and the created database is used in training and testing the artificial neural network model. The performance of the proposed artificial neural network (ANN) is determined using ambient vibration tests conducted on both undamaged buildings from the literature and damaged buildings during the Samos earthquake (2020) in the scope of this study. As a result, it has been shown that the proposed ANN is quite successful and can be used as an alternative method for determining the period values of undamaged—damaged RC buildings without the need to generate complex 3D numerical models.