Twistor spaces of K3 surfaces are non-Kähler compact complex manifolds which play a fundamental role in the moduli theory of K3 surfaces. They come equipped with a holomorphic submersion to the complex projective line which under the period map corresponds to a twistor line in the K3-period domain. In this talk I will explain how one can view a twistor line as a certain base point in the linear cycle space of the period domain. Then, based on joint work in progress with Daniel Greb, Tim Kirschner and Martin Schwald I will present new results concerning the deformations of twistor spaces of K3 surfaces and their relation to the cycle space of the period domain.