We study Vassiliev invariants of links in a 3-manifold M by using chord diagrams labeled by elements of the fundamental group of M. We construct universal Vassiliev invariants of links in M, where M=P2×[0,1] is a cylinder over the real projective plane P2, M=Σ×[0,1] is a cylinder over a surface Σ with boundary, and M=S1×S2. A finite covering p:N⟶M induces a map π1(p)∗ between labeled chord diagrams that corresponds to taking the preimage p−1(L)⊂N of a link L⊂M. The maps p−1 and π1(p)∗ intertwine the constructed universal Vassiliev invariants.