A new inverse technique to characterise the nonlinear mechanical properties of viscoelastic core layers in sandwich panels has been developed based on simple vibration tests. The present methodology allows one to preserve the frequency and temperature dependences of the storage and loss moduli of viscoelastic materials for a wide range of frequencies and to perform a structural analysis using high damping tests. The computational effort has been substantially reduced by using an optimisation based on the planning of the experiments and the response surface technique in order to minimise the error functional. This new inverse technique has been tested and applied to characterise the viscoelastic properties of a 3M damping polymer (ISD-112) used as a core material in sandwich panels. The identified material properties have been verified successfully by comparing structural dynamic parameters obtained from the numerical analysis with experiments.