The contribution to the nuclear transverse response function R_T arising from two particle-two hole (2p-2h) states excited through the action of electromagnetic meson exchange currents (MEC) is computed in a fully relativistic framework. The MEC considered are those carried by the pion and by Delta degrees of freedom, the latter being viewed as a virtual nucleonic resonance. The calculation is performed in the relativistic Fermi gas model in which Lorentz covariance can be maintained. All 2p-2h many-body diagrams containing two pionic lines that contribute to R_T are taken into account and the relative impact of the various components of the MEC on R_T is addressed. The non-relativistic limit of the MEC contributions is also discussed and compared with the relativistic results to explore the role played by relativity in obtaining the 2p-2h nuclear response.