Using of the multiple scattering methods, we characterize the positive and negative multi-refraction and transmission properties of a solid-based phononic crystal composed of coated solid inclusions in view of its applications in tunable multimode filtering. The geometrical parameters are chosen so that a left-handed longitudinal wave mode and a right-handed transverse wave mode, are simultaneously obtained in this three-component phononic crystal. When multimode Gaussian beams are placed transmitting through the phononic crystal slab, both positive and negative refractions are observed. We then study the individual propagation behavior of different modes. The angle dependent transmission beams with different energy distributions are found at the other side of the slab. Transmitted transverse waves coming from different directions incidence finally walk together into four oriented beams. Meanwhile, longitudinal wave incidence with different directions behaves simply as negative refraction in the slab. A far-field longitudinal wave image can be achieved being excited by a longitudinal wave point source. The three-component phononic crystal slab thus can be served as an alternate in tunable multimode filtering devices.