Pakistan's army said the attacks in Kurram tribal district on Sunday also wounded three soldiers during the search for those who held American Caitlan Coleman, her Canadian husband, Joshua Boyle, and their three children hostage.
The family were freed on Wednesday when the Pakistani army shot out the tyres of a vehicle carrying the family during a rescue based on intelligence shared by US authorities.
A local government official, Baseer Khan, said an improvised explosive device exploded when a military bomb disposal squad was scanning the route, and the other two bombs went off when an army team reached the site.
The Pakistani Taliban claimed responsibility.
"These personnel were searching for the kidnappers of a US citizen and her family," said David Hale, US Ambassador to Pakistan, in a statement.
"We remain extremely grateful for the Pakistani military's quick response and successful humanitarian operation allowing Caitlan Coleman and her family to return home safely."
The family's rescue has been hailed by US President Donald Trump as a "positive moment" for US-Pakistan relations, which have frayed in recent years amid Washington's assertions that Islamabad has not been doing enough to tackle Taliban-linked Haqqani militants who are believed to be on Pakistani soil.
Coleman and Boyle were held by Haqqani militants who kidnapped them while backpacking in Afghanistan in 2012.
The Pakistani army has indicated that the captors were tracked shortly after entering from Afghanistan, although it remains unclear whether the family were kept in Afghanistan for all five years, or in Pakistan for some of the time.
Haqqani terrorists, once termed by a US general a veritable arm of Pakistan's top spy agency, the Inter Service Intelligence (ISI), operate on both sides of the long porous border.
Boyle, in a video statement released by the Pakistani military, called his captors criminals and pagans who had nothing to do with Islam.
Boyle described the operation to free his family as "incredibly" professional.
"I did see the truth, and the truth was that car was riddled with bullets," he said. "The ISI and the army got between the criminals and that car to make sure that the prisoners were safe, my family was safe."
(This story has not been edited by NDTV staff and is auto-generated from a syndicated feed.)