We investigate the diffusion of cosmic rays (CRs) close to their sources. Propagating individual CRs in purely isotropic turbulent magnetic fields with maximal scale of spatial variations l(max), we find that CRs diffuse anisotropically at distances r ≤/~ l(max) from their sources. As a result, the CR densities around the sources are strongly irregular and show filamentary structures. We determine the transition time t(*) to standard diffusion as t(*) ~ 10(4) yr(l(max)/150 pc)(β)(E/PeV)(-γ)(B(rms)/4 μG)(γ), with β =/~ 2 and γ=0.25-0.5 for a turbulent field with a Kolmogorov power spectrum. We calculate the photon emission due to CR interactions with gas and the resulting irregular source images.