Motivated by recent experiments, we analyze the stability of a three-dimensional Bose-Einstein condensate loaded in a periodically driven one-dimensional optical lattice. Such periodically driven systems do not have a thermodynamic ground state but may have a long-lived steady state which is an eigenstate of a "Floquet Hamiltonian." We explore collisional instabilities of the Floquet ground state which transfer energy into the transverse modes. We calculate decay rates, finding that the lifetime scales as the inverse square of the scattering length and inverse of the peak three-dimensional density. These rates can be controlled by adding additional transverse potentials. © 2015 American Physical Society.