The design of the Linear Gridless Ion Thruster, or LGIT, is presented in detail. The LGIT is a hybrid thruster that combines an ionization stage similar to those normally found on gridded ion engines with the acceleration mechanism of a Hall effect thruster. This thruster also features a linear geometry, which simplifies the design of the magnetic circuit while making the LGIT particularly well suited to future work on clustering and, perhaps, thrust vectoring by varying the magnetic fields in the acceleration zone. Initial testing with the thruster operating in a single-stage, unoptimized mode resulted in a specific impulse of 1400 seconds and an anode efficiency of 12%. The low efficiency is believed to be due in large part to operating the thruster in a single-stage mode, rather than the two-stage mode for which it was designed, and to setting the acceleration stage magnets to less than half their design value.