This paper addresses the problem of determining the relaxation spectrum of a viscoelastic fluid from knowledge of the phase velocity of shear waves propagated in the fluid. Theoretical constraints are obtained which guarantee solvability of the problem. Questions of ill-posedness are identified and it is shown how to include additional information into the problem from independent measurements of group velocity. Alternative integro-differential and integro-algebraic systems are derived which are closed with respect to the original and additional data. A simple iterative algorithm which involves updating estimates of a moment ratio is proposed and tested against two model problems. The algorithm determines an n-mode relaxation spectrum and the corresponding storage and loss moduli. (C) 2004 Elsevier B.V. All rights reserved.