B767-400 First Officer reported they lost all radios and returned to their departure airport.

Narrative

We received the takeoff clearance from tower. After we reached 3000 feet; I queried ATC for a handover to departure. I received no response so I tried again; with no response. The Captain said he did not hear my transmission. I keyed the hand microphone but still could not get a response from ATC. The frequency was completely silent. We then realized all three audio control panels were blank. The Captain then turned toward our first fix as per lost comm procedures. We then had a discussion about what altitude we should fly. I knew that 10000 feet was a very common assignment by departure so we climbed to 10000 feet. After we reached 10000 feet the Captain transferred controls to me. I made the recommendation that we continue the climb to the altitude told to expect in our clearance which was 31000 feet; as per lost comm procedures. We then started a climb to 31000 ft. and then received a message from dispatch that ATC had called and wanted us to stay at 10000 feet. I was passing 10200 at this time so I descended back down to 10000 ft. We looked in the QRH for a loss of audio control panel function but found nothing. Dispatch also sent a routing from ATC which included a holding pattern over ZZZ [VOR] and to let them know when we wanted to come back to ZZZ. The dispatcher tried to call over SATCOM; but since the audio control panels were all dead; there was no way to answer the call. The Captain pulled circuit breakers related to the COMM system; and reset them but this did not solve the issue. A request was made through dispatch to contact maintenance for any suggestions. They were unable to find a solution. Dispatch advised we could burn off the gas needed to land underweight or do an overweight landing. We ran the overweight landing checklist. First officer had the idea to turn on his cellphone and try to get a signal which worked and the Captain was able to call the dispatcher and talk with maintenance. Maintenance still had no suggestions to try and fix the audio panels. Dispatch then provided 2 phone numbers; one for ZZZ tower and one for approach. First officer called approach with the cellphone and was able to get instructions on when to leave the hold and proceed on course to begin the ILS approach to runway XR. He said to look for the light gun signal from tower and after landing to taxi to the gate. The Captain relayed that he wanted a follow me truck waiting after we landed to guide us to the gate. We did an arrival briefing; left the hold and proceeded toward ZZZ1 [VOR] then ZZZZZ and intercepted the ILS. Around ZZZZZ1 we received a green light from the tower; performed a landing where an OPS vehicle was waiting and we followed it to gate. The passengers and flight attendants were kept informed during the whole event. Maintenance found one loose plug and one plug that was barely plugged in; in the electronics bay.