The integration of FGMs in curved beam design offers considerable potential. These materials are intentionally engineered to exhibit continuous variations in properties along specific directions, granting designers the flexibility to tailor strength and stiffness distributions to meet structural demands. This paper investigates the free vibration behavior of circular beams whose material properties vary continuously through the beam thickness. Specifically, the studied beam features a variation in Young’s modulus and density along the height of its cross-section, following a symmetric distribution with respect to the centroid. The exact dynamic stiffness matrix (DSM) for a single circular beam element is derived in a suitable dimensionless form, which facilitates the assembly of the global stiffness matrix for various boundary conditions. Natural frequencies and corresponding vibration modes are determined using an analytical approach based on the properties of the dynamic stiffness matrix, employing the Wittrick–Williams algorithm. The DSM yields exact results for all natural frequencies and mode shapes without resorting to any approximation. Numerical results are presented to demonstrate the advantages of the proposed method and to validate the derived exact closed-form solution. For arches with varying support conditions, the influence of geometrical parameters and of the ratio between the mechanical properties of the inner and outer materials on the natural frequencies and associated vibration modes is also analyzed.