We present measurements of the redshift-space three-point correlationfunction of 50967 luminous red galaxies (LRGs) from Data Release 3 (DR3)of the Sloan Digital Sky Survey (SDSS). We have studied the shapedependence of the reduced three-point correlation function(Qz(s, q, θ)) on three different scales, s = 4, 7 and10h-1Mpc, and over the range of 1 < q < 3 and 0°< θ < 180°. On small scales (s = 4h-1Mpc),Qz is nearly constant, with little change as a function of qand θ. However, there is evidence for a shallow U-shapedbehaviour (with θ) which is expected from theoretical modellingof Qz(s, q, θ). On larger scales (s = 7 and10h-1Mpc), the U-shaped anisotropy in Qz (withθ) is more clearly detected. We compare this shape dependence inQz(s, q, θ) with that seen in mock galaxy catalogueswhich were generated by populating the dark matter haloes in largeN-body simulations with mock galaxies using various halo occupationdistributions (HOD). We find that the combination of the observed numberdensity of LRGs, the (redshift-space) two-point correlation function andQz(s, q, θ) provides a strong constraint on theallowed HOD parameters (Mmin, M1,α) andbreaks key degeneracies between these parameters. For example, ourobserved Qz(s, q, θ) disfavours mock catalogues thatoverpopulate massive dark matter haloes with many LRG satellites. Wealso estimate the linear bias of LRGs to be b = 1.87 +/- 0.07 inexcellent agreement with other measurements.