In contrast to the Miles–Howard theorem for inviscid steady shear flow in stably stratified fluids, explicit elementary time-periodic solutions of the Boussinesq equations are developed here which are unstable for arbitrarily large Richardson numbers. These elementary flows are parameterized through solutions of a nonlinear pendulum equation and involve spatially constant but temporally varying vorticity and density gradients which interact through advection and baroclinic vorticity production. Exact nonlinear solutions for arbitrary wave-like disturbances for these flows are developed here and Floquet theory combined with elementary numerical calculations is utilized to demonstrate instability at all large Richardson numbers. The dominant inviscid instability for these non-parallel flows is a purely two-dimensional parametric instability with twice the period of the elementary flow and persists for all Reynolds numbers and the wide range of Prandtl numbers, 1[les ]Pr[les ]200, investigated here. Similar elementary time-periodic solutions of the Boussinesq equations in a constant external strain field are developed here which reduce to uniform shear flows in one extreme limit and the time-periodic vortical flows in the other extreme limit. These flows are stable in a strict sense for large Richardson numbers; however there is transient large-amplitude non-normal behaviour which yields effective instability for a wide range of Richardson numbers. For example, suitable initial perturbations can amplify by at least a factor of fifty with exponential growth for short times for [Rscr ]i=1, [mid ]σ[mid ][les ]0.5 and [Rscr ]i=5, [mid ]σ[mid ][les ]0.1, with σ the amplitude of the external strain.