Rigid scatterers are fundamentally important in the study of scattering of many types of waves. However, in the recent literature on scattering of flexural waves on thin plates, a “rigid scatterer” has been used to represent a clamped boundary. Such a model physically resembles riveting the plate to a fixed structure. In this paper, a movable model for a rigid scatterer that allows rigid-body motion is established. It is shown that, when the mass density of the movable rigid scatterer is much greater than that of the host plate and at high frequencies, the movable rigid scatterer approaches the limiting case that is the riveted rigid scatterer. The single- and multiple-scattering by such scatterers are examined. Numerical examples show that, at the extreme end of lower frequencies, the scattering cross section for the movable model vanishes while that of the riveted models approaches infinity. An array of such movable rigid scatterers can form a broad and well-defined stop band for flexural wave transmission. With a volume fraction above 50%, the spectrum is rather clean: consisting of only an extremely broad stop band and two groups of higher frequency Perot–Fabry resonance peaks. Increasing either scatterer’s mass density or the lattice spacing can compress the spectral features toward lower frequencies.