Two of the container trucks caught fire and burned from the grenade blasts at about 2:30 a.m. on the main highway about 60 kilometers (37 miles) east of Kabul, said Sarhadi Zwak, spokesman for Laghman province where the attack took place.
It was unclear if the trucks were carrying supplies for the U.S.-led military coalition or if they had commercial goods, Zwak said, but he added that no international troops or military vehicles were in the convoy. He said the drivers were Afghan.
Attacks on supply trucks coming to Kabul are relatively common, as Taliban and other militants seek to disrupt both flow of the military goods and sow fear among the population.
The latest attack comes in a period of intense violence, with the Taliban targeting police and civilian officials and attacking government positions around the country as Afghan police and army prepare to officially take over full responsibility for security from international troops.
Insurgents are also waging a campaign of assassinations of government officials. On Monday, the police chief of the southern province of Helmand survived a suicide car bomb attack on his convoy that wounded three officers.
Police Chief Mohammad Nabi Elham sustained only minor injuries when the car bomber struck as he was on his way to his office at about 7 a.m.
Broken glass and the charred remains of the bomber's car were strewn in a main road in the provincial capital of Lashkar Gah after the attack. The force of the blast tore off the door of Elham's vehicle. Three police officers traveling in the convoy were wounded, provincial spokesman Ummar Zawaq said.
"Thanks be to God that it was so early in the morning," Elham said later in an interview. "If it had been 8 or 9 in the morning, there would have been laborers here who are building a road for a mosque. Shopkeepers would have been here, and how many people might have been killed?"
With Afghans for the first time taking the lead in fighting the Taliban in most of the country this year, the toll on Afghan forces has been high, more than doubling from last year's spring and summer fighting season. In May alone, at least 271 police were killed in attacks, and total deaths for all security forces including the army and community-based forces known as the local police was 400 for the month.
At the same time, casualties among the U.S.-led military coalition have been reducing as the international forces pull back to let the Afghans take the lead. In May, 21 NATO troops were killed in the country, down from 44 during the same month last year.
The coalition said that one of its service members died in a non-battle-related incident in southern Afghanistan on Sunday but released no further details. The death brings June's toll for international troops to 20.
Associated Press writer Mirwais Khan in Kandahar, Afghanistan contributed to this report.