  A number of merging galaxy clusters show the presence of large-scale radio
emission associated with the intra-cluster medium (ICM). These synchrotron
sources are generally classified as radio haloes and radio relics. Whilst it is
commonly accepted that mergers play a crucial role in the formation of radio
haloes and relics, not all the merging clusters show the presence of giant
diffuse radio sources and this provides important information concerning
current models. The Abell 781 complex is a spectacular system composed of an
apparent chain of clusters on the sky. Its main component is undergoing a
merger and hosts peripheral emission that is classified as a candidate radio
relic and a disputed radio halo. We used new LOw Frequency ARay (LOFAR)
observations at 143 MHz and archival Giant Metrewave Radio Telescope (GMRT)
observations at 325 and 610 MHz to study radio emission from non-thermal
components in the ICM of Abell 781. Complementary information came from
XMM-Newton data, which allowed us to investigate the connection with the
thermal emission and its complex morphology. The origin of the peripheral
emission is still uncertain. We speculate that it is related to the interaction
between a head tail radio galaxy and shock. However, the current data allow us
only to set an upper limit of $\mathcal{M} < 1.4$ on the Mach number of this
putative shock. Instead, we successfully characterise the surface brightness
and temperature jumps of a shock and two cold fronts in the main cluster
component of Abell 781. Their positions suggest that the merger is involving
three substructures. We do not find any evidence for a radio halo either at the
centre of this system or in the other clusters of the chain. We place an upper
limit to the diffuse radio emission in the main cluster of Abell 781 that is a
factor of 2 below the current radio power-mass relation for giant radio haloes.
