A method of fundamental solutions (MFS) is presented for the ill-posed linear inverse problem consisting of the reconstruction of boundary data on the inner boundary for the hyperbolic system of elastodynamics in planar annular domains from known essential and natural boundary conditions on the outer boundary. This corresponds to the problem of finding elastic wave propagation in a structure from measured data being the displacement and traction on a portion of the boundary of the structure. The time-dependent lateral Cauchy problem is reduced to a sequence of elliptic systems by applying the Laguerre transform. A sequence of fundamental solutions to the elliptic equations are derived. Linear combination of elements of this sequence of fundamental solutions is used to generate an approximation to the elliptic Cauchy problems. By placing source points outside of the solution domain, and collocating on the boundary, linear equations are obtained for finding the coefficients in the MFS approximation. It is outlined that the sequence of fundamental solutions of the elliptic systems constitutes a linearly independent and dense set on the boundary with respect to the L-2-norm. Tikhonov regularization in combination with the L-curve rule is incorporated to generate a stable solution to the obtained systems of linear equations. The proposed MFS approximation for the time-dependent lateral Cauchy problem is supported by numerical results.