Major James Joshua Bowman, from Wiltshire, Lieutenant Neal Turkington, from Craigavon, and Corporal Arjun Purja Pun, from Nepal, served in 1st Army The Royal Gurkha Rifles.
The Ministry of Defence appear their names as ancestors associates and assembly said they were "selfless" and "brave".
A above manhunt is beneath way afterward Tuesday's advance in Nahr-e Saraj.
Maj Bowman was attempt as he slept, while Lt Turkington and Corporal Pun died back a rocket-propelled armament was accursed into the command centre.
Maj Bowman's ancestors said he had been the "best accessible son and brother" who "loved the Army and was actual appreciative of the affectionate assignment that he and his aggregation were doing".
Lt Turkington's ancestors said they were "devastated" by their accident and said the soldier had been "relentless and abiding in his following of those causes he believed in".
Cpl Pun's wife, Durga, said her bedmate had been "proud to be a soldier and died accomplishing a job he loved".
Lt Col Gerald Strickland MBE, Commanding Administrator 1st Army The Royal Gurkha Rifles, said that with Above Bowman's death, the army had absent a "brave" and "truly courageous" administrator who had led with a "rare determination".
The 34-year-old from Salisbury had abutting The Light Infantry in 1999 and served in Northern Ireland and Sierra Leone afore abutting 1st Army The Royal Gurkha Rifles in May 2009.
Lt Col Strickland said Lt Turkington, 26, had been a "courageous and bent army commander" who was admired for his "integrity and raw ability".
Lt Col Strickland said Cpl Pun, 33, a father-of-two from Khibang apple in Nepal's Magdi District, had been a "true Gurkha... abiding and loyal", and a man who had "immediately becoming the account of his men".
'Not forgotten'
Earlier, Nato said it was application every "asset" aural its ability to acquisition the Afghan analgesic and those who may be acceptable him.
Defence Secretary Liam Fox said the "despicable and treacherous" act would "not cavity the boldness of the colleagues they leave behind".
He said: "The efforts of 1st Army the Royal Gurkha Rifles accept anon bigger the aegis bearings in the Nahr-e Saraj district; their adventurous acts accept bigger the lives of the Afghan bodies and their mission to accomplice and alternation the Afghan National Army is basic to our success.
"I extend my centermost condolences to the families and admired ones of the servicemen asleep in this awful attack, their cede will not be forgotten."
A Royal Marine was additionally asleep on Tuesday back he was attempt asleep in a abstracted adventure in Helmand province, demography the cardinal of British cadre asleep in Afghanistan back 2001 to 318. He has not yet been named.
On Wednesday, the Ministry of Defence appear that Gen Sir David Richards is to be the abutting arch of the defence staff, the UK's best chief aggressive officer.
He will alter Air Arch Marshal Sir Jock Stirrup, who is abrogation the column after this year.
