We present here the design and the optimization of a dual-band dual linearly-polarized transmitarray antenna with a 1-bit phase resolution (i.e. two phase states with 180° of relative phase-shift). The antenna operates in both downlink and uplink K/Ka-bands (17-21 GHz for the lower frequency band, and 27-31 GHz for the upper one) and could be used for satellite communication applications. In order to implement dual-band and dual-polarization, the unit-cell consists of four orthogonal superposed U-slotted patch antennas interconnected by a metallized via hole. A 20x20 transmitarray based on the proposed unit-cell architecture has been optimized by using our in-house tool and validated through full-wave electromagnetic simulations. The 5x5 mm² unit-cell size allows beam scanning over large field of view (up to 80 deg.). A maximum gain of 23.4 and 21.4 dBi has been achieved at 29.1 GHz and 19.5 GHz, respectively. The gain 1-dB bandwidth is equal to 17.4% and 10.5% at the lower and the upper frequency bands, respectively.