The problem of two-dimensional acoustic scattering of time-harmonic plane waves by a multi-ringed cylindrical resonator is considered. The resonator is made up of an arbitrary number of concentric sound-hard split rings with zero thickness. Each ring opening is oriented in any direction. The acoustics pressure field in each layered region enclosed between adjacent rings is described by an eigenfunction expansion in polar coordinates. An integral equation/Galerkin method is used to relate the unknown coefficients of the expansions between adjacent regions separated by a ring. The multiple scattering problem is then formulated as a reflection/transmission problem between the layers, which is solved using an efficient iterative scheme. An exploration of the parameter space is conducted to determine first, the conditions under which the lowest resonant frequency can be minimised, and second, how non-trivial resonances of the multi-ring resonators can be explained from those of simpler arrangements, such as a single-ring resonator. It is found here that increasing the number of rings while alternating the orientation lowers the first resonant frequency, and exhibits a dense and nearly regular resonant structure that is analogous to the rainbow trapping effect.