Tunneling spectra obtained from [Al-AlOx-pA-Pb] (where pA = anthracene, tetracene, pentacene, perylene, and coronene) tunnel diodes are reported in the bias region extending from -2.3 to +2.3 V. Large intense features (delta sigma/sigma similar to 0.1) are seen in the spectrum when Pb is biased positively but are absent in the other bias. These features are not scan direction dependent and do not correlate with inelastic electronic transitions. The positions of the band maxima vary linearly with the Huckel LUMO energy and can be fit using a value of beta = 67 kcal/mol. These novel bands are attributed to elastic tunneling mediated by the unoccupied pi* orbitals of the polyacenes. While this type of tunneling has been previously postulated for the case of copper phthalocyanine, we believe that the present study is the first unequivocal demonstration of resonance-like elastic tunneling via unoccupied molecular orbitals. The implications of these results to molecular imaging in scanning tunneling microscopy are also discussed.