Unconventional methods are needed in order to propel swimming objects in viscoelastic liquids. The paper deals with a locomotion principle based on a cyclic action which utilizes the elasticity of the fluid. The theoretical model considers any incompressible simple fluid of arbitrarily long memory and any axisymmetric swimmer of arbitrary profile which performs torsional oscillations of small amplitude. Perceiving the flow as unsteady perturbation of the rest state, an asymptotic analysis is developed, particularly with regard to the time-averaged speed of the swimmer in second-order approximation. In doing so, also inertia effects are considered in addition to the memory and the normal stress effects. A generalized reciprocal theorem including fluid elasticity proves to be extremely useful. It enables calculating the driving force on the swimmer without solving the second-order flow problem. General results are illustrated by means of a spherical swimmer. The analytical findings clearly show the influence of different process parameters including certain frequency dependent constitutive parameters on the driving force, on the swimming speed and on the secondary flow field. (C) 2015 Elsevier B.V. All rights reserved.