Optical turbulence in driven nonlinear resonators is a form of spatiotemporal chaos that remains poorly understood and is largely characterized through simulations and experiments but lacks a dynamical-systems description. Here we show that in the regime of weak spatiotemporal chaos, the dynamics is structured by families of exact, fully nonlinear, time-periodic solutions of the Lugiato-Lefever equation for driven-dissipative Kerr resonators. These unstable periodic orbits, which we term Oscillating Turing Rolls (OTR), are embedded within the spatiotemporal chaos of optical turbulence. Parametric continuation using Newton-Krylov methods reveals their existence range, stability, and bifurcation-theoretic origin. In symmetry-constrained domains, OTR solutions are intermittently shadowed by chaotic trajectories; in extended domains, they appear as localized motifs of spatiotemporal chaos. Identifying these nonlinear, non-chaotic building blocks establishes a dynamical-systems-based description of optical turbulence in nonlinear optical resonators.