Optical absorption spectra have been calculated for a series of boron nitride fullerenelike cage structures BnNn of sizes n=12-36. The method used is a real-time, real-space implementation of the time-dependent density-functional theory, involving the full time propagation of the time-dependent Kohn-Sham equations. The spectra are found to be a possible tool for distinguishing between different boron nitride fullerene species and isomers. The trends and differences in the spectra are found to be related to the general geometry of the molecules. Comparison between local-density and generalized-gradient approximations for electron exchange-correlation functionals shows that both of them produce essentially the same spectral characteristics.