The effect of molecular structure and orientation on the rate of tunnel ionization in intense laser fields is considered for linear molecules. The nonspherical nature of the molecular electrostatic potential near the classical electron turning point results in a nonisotropic distribution of tunneling rates. This must be considered when determining the overall ionization rate of a randomly oriented ensemble. Calculations are performed for two atom:molecule test pairs with similar ionization potential, Ar:N-2 and Xe:O-2, and the results are compared to experiments [Talebpour, Chien, and Chin, J. Phys. B 29, L677 (1996); Guo , Phys. Rev. A 58, R4271 (1998)].