This paper examines the effect of boundary conditions on the band-gap properties of flexural waves in a periodic compound plate. The general boundary conditions are modelled by linear and torsional springs, and the traditional free, clamped, and simply supported boundary conditions become their special cases when the spring constants approach extreme values. The forced response of a finite periodic structure and the band-gap frequencies of an infinite periodic structure are solved analytically using the thin plate equations with the boundary conditions and Bloch periodic conditions. The results show that the band-gap and propagating mode properties of the compound plate are highly dependent on the boundary stiffness constants. For a small stiffness, a few branches of dispersion curves tend to be concentrated in the same pass-band. They gradually become separated as the stiffness increases, resulting in band-gaps with broader width. In addition, in the frequency range of interest, the band-gap properties are more sensitive to linear spring stiffness than torsional spring stiffness. The linear spring stiffness has a significant influence on all the dispersion curves, while the effect of the torsional spring stiffness on the band-gap properties varies with the dispersion branches.