The problem of a vibrating-rotating polyatomic molecule is treated, with emphasis given to the case of molecules with C_3v symmetry. It is shown that several of the gross features of the rotational spectra of polyatomic molecules in excited vibrational states can be predicted by group theoretical considerations. Expressions for the rotational transition frequencies of molecules of C_3v symmetry in the ground vibrational state, singly excited degenerate vibrational states, and doubly excited degenerate vibrational states are given. The derivation of these expressions by fourth order perturbation theory as given by Amat, Nielsen, and Tarrago is discussed. The ground and V_10=1 rotational spectra of propyne have been investigated in the 17 to 70 GHz, and 17 to 53 GHz regions, respectively, and compared with predictions based on higher frequency measurements. The V_9=1 and V_10=2 rotational spectra of propyne have been investigated and assigned for the first time. A perturbation of the V_9=1 rotational spectra for K=-l has been discovered and discussed.