Turbulent vortex dynamics is investigated in triply periodic turbulent flow with Kida’s high symmetry (Kida, J. Phys. Soc. Japan, vol. 54, 1985, pp. 2132–2136) by means of unstable periodic motion representing both the statistical and dynamical properties of turbulence (van Veen et al., Fluid Dyn. Res., vol. 38, 2006, pp. 19–46). In the periodic motion, the large-scale columnar vortices, the smaller-scale vortices and the large-amplitude axial waves on the large-scale columnar vortices are detected. In terms of mutual dynamical interaction between the large-scale columnar vortices and smaller-scale vortices, we demonstrate a cyclic process of excitation of the axial waves, which leads to large-amplitude fluctuations of the total kinetic energy and enstrophy. This cyclic process is characterised by three distinct phases and is therefore reminiscent of the regeneration cycle of near-wall turbulence structures (Hamilton et al., J. Fluid Mech., vol. 287, 1995, pp. 317–348). Notably, such oscillatory behaviour is observed even in freely decaying turbulence as a consequence of the instantaneous energy transfer from smaller to larger scales.