A new experimental investigation of the level scheme of the N=Z+1 nucleus Zr-81 is reported. An additional band assigned to the [431]1/2(+) orbital has been observed and the known [422]5/2(+) and [301]3/2(-) bands have been extended to higher spins. The behavior of these bands is compared to that of the bands in the isotonic nucleus Sr-79, for which some new experimental data are presented. Comparison is also made to the even-even N=Z and N=Z+2 neighbors. The band structures are discussed in the framework of the projected shell model.