Magnetic skyrmions are vortex-like spin textures, which are usually treated as two-dimensional objects. In their lattice state, they form well-ordered, hexagonal structures, which have been studied in great detail. To obtain a three-dimensional (3D) skyrmion crystal, these planes can be envisaged to be stacked up forming skyrmion strings in the third dimension. Here, we report the observation of a 3D skyrmion phase in Cu2OSeO3 by carrying out reciprocal space mapping in resonant elastic x-ray scattering. We observe regions in the magnetic field-cooling phase diagram in which the skyrmion phase apparently coexists with the conical phase. However, such a coexistence is forbidden due to symmetry arguments. Instead, the skyrmion strings themselves are periodically modulated along their axes, as confirmed by micromagnetic simulations. The periodic modulation is in fact a necessary consequence of the evolution of the skyrmion phase out of the conical state and should therefore be a universal property of skyrmion strings in chiral helimagnets.