To study the elastic properties of rodlike DNA nanostructures, we perform long simulations of these structures using the oxDNA coarse-grained model. By analyzing the fluctuations in these trajectories, we obtain estimates of the bend and twist persistence lengths and the underlying bend and twist elastic moduli and couplings between them. Only on length scales beyond those associated with the spacings between the interhelix crossovers do the bending fluctuations behave like those of a wormlike chain. The obtained bending persistence lengths are much larger than that for double-stranded DNA and increase nonlinearly with the number of helices, whereas the twist moduli increase approximately linearly. To within the numerical error in our data, the twist-bend coupling constants are of order zero. That the bending persistence lengths that we obtain are generally somewhat higher than in experiment probably reflects both that the simulated origamis have no assembly defects and that the oxDNA extensional modulus for double-stranded DNA is too large.