  We investigate the spin dynamics of excitons interacting with acoustic
phonons in quantum wells, quantum wires and quantum disks by employing a
multiband model based on the $4\times4$ Luttinger Hamiltonian. We also use the
Bir-Pikus Hamiltonian to model the coupling of excitons to both longitudinal
acoustic phonons and transverse acoustic phonons, thereby providing us with a
realistic framework in which to determine details of the spin dynamics of
excitons. We use a fractional dimensional formulation to model the excitonic
wavefunctions and we demonstrate explicitly the decrease of spin relaxation
time with dimensionality. Our numerical results are consistent with
experimental results of spin relaxation times for various configurations of the
GaAs/Al$_{0.3}$Ga$_{0.7}$As material system. We find that longitudinal and
transverse acoustic phonons are equally significant in processes of exciton
spin relaxations involving acoustic phonons.
