Twisted nematic liquid crystal displays (TN-LCD) are commercially available and have found a widespread use for displaying dynamic diffractive elements. Usually these devices are inserted between two polarizers but then a strong amplitude-phase coupling is obtained. In this paper we show that by using elliptically polarized light at the entrance and detecting elliptically polarized light at the output, it is possible to obtain a pure phase-only modulation. The complete device then consists on a polarizer, a waveplate, the TN-LCD, another waveplate and another polarizer. This technique is particularly useful for newer LCDs which are thinner devices and require short wavelengths to achieve a 2π phase modulation depth. If a phase-only diffractive element is displayed by an LCD which does not produce this perfect phase-only modulation, different diffraction orders are generated. We include a model to evaluate the diffraction efficiency, which accounts for a phase modulation depth less than 2π, non-linearities in the phase modulation or a coupled amplitude modulation. We derive the expression to evaluate the amplitude of these diffraction orders which are related with the light efficiency that can be obtained. It is demonstrated that a good efficiency can be obtained even though the maximum phase modulation depth is less than 2π. For instance, this is the situation if a thin display is illuminated with a long wavelength. The diffraction efficiency is higher than 90% when the modulation depth is 1.4π, and higher than 40% (the equivalent for a binary phase element) when the modulation depth is 0.7π.