We identify a quasi-two-dimensional (quasi-2D) phonon mode in the layered-perovskite Ca_{3}Ti_{2}O_{7}, which exhibits an acoustic branch with quadratic dispersion. Using first-principles methods, we show this mode exhibits atomic displacements perpendicular to the layered [CaTiO_{3}]_{2} blocks comprising the structure and a negative Grüneisen parameter. Owing to these quasi-2D structural and dynamical features, we find that the mode can be utilized to realize unusual membrane effects, including a tunable negative thermal expansion (NTE) and a rare pressure-independent thermal softening of the bulk modulus. Detailed microscopic analysis shows that the NTE relies on strong intralayer Ti-O covalent bonding and weaker interlayer interactions, which is in contrast to conventional NTE mechanisms for perovskites, such as rigid-unit modes, structural transitions, and electronic or magnetic ordering. The general application of the quasi-2D lattice dynamics opens exciting avenues for the control of lattice dynamical and thermodynamic responses of other complex layered compounds through rational chemical substitution, as we show in A_{3}Zr_{2}O_{7} (A=Ca, Sr), and by heterostructuring.