The Cryogenic Underground Observatory for Rare Events (CUORE) at Laboratori Nazionali del Gran Sasso of INFN in Italy is an experiment searching for neutrinoless double beta (0νββ) decay. Its main goal is to investigate this decay in 130Te, but its ton-scale mass and low background make CUORE sensitive to other rare processes as well. In this Letter, we present our first results on the search for 0νββ decay of 128Te, the Te isotope with the second highest natural isotopic abundance. We find no evidence for this decay, and using a Bayesian analysis we set a lower limit on the 128Te 0νββ decay half-life of T1/2>3.6×1024  yr (90% CI). This represents the most stringent limit on the half-life of this isotope, improving by over a factor of 30 the previous direct search results, and exceeding those from geochemical experiments for the first time.