A robust inverse method for the complex wavenumber space (complex k-space) extraction is essential for structural vibration and damping analysis of two-dimensional structures. Most existing methods suffer from extracting the reliable complex k-space of plates in the presence of realistic uncertainties, especially for plates with low damping properties. To this end, this paper presents a new method for extracting the dispersion and damping characteristics of two-dimensional periodic structures using only the full-field displacement fields as input. The proposed method, the Algebraic K-Space Identification 2D technique (AKSI 2D), is an extension of the Algebraic Wavenumber Identification technique to solve two-dimensional problems. The optimised formulas are developed within the algebraic identification framework, which allows the extraction of all the properties of the complex k-space in a comprehensive way. The proposed method is validated numerically and experimentally, and its performances are compared with other popular k-space identification methods under different uncertainty conditions. The test cases cover analytically solved isotropic fields to numerically solve orthotropic fields and finally experimental measurements. The different cases show promising results and demonstrate that the proposed method is a robust tool to characterise the wave propagation of two-dimensional structures under stochastic structural and constitution conditions.