Gamma-ray transitions in the extremely neutron-deficient nucleus Te-106 have been identified for the first time. The experiment utilized the Fe-54(Fe-54,2n)Te-106(*) reaction, and the gamma-ray transitions from excited states in Te-106 were selected by use of the recoil-decay-tagging technique. The production cross section was estimated at 25 nb, a new limit for in-beam gamma-ray spectroscopy. A ground-state band tentatively extending up to I-pi=10(+) is proposed. The systematics of low-lying yrast states in the Te isotopes is discussed within the context of vibrational excitations and residual nucleon-nucleon interactions.