We study unstable, time-periodic solutions in Large Eddy Simulation (LES) of Homogeneous, Isotropic Turbulence (HIT). The turbulence is forced at large spatial scales by an external body force which induces four vortex columns in a rectangular domain with periodic boundary conditions. The dissipation range dynamics is represented by the Smagorinsky model. Both the kinematic viscosity and the Smagorinsky constant are used as a homotopy parameters connecting steady, laminar flow to turbulent inertial range dynamics. We find several families of periodic solutions, some bifurcating from the steady flow and some filtered from more turbulent dynamics, and attempt to track these solutions to the limit of fully developed turbulence.