One of the places is Samangan province in the north, where the Taliban said that it attacked NATO supply trucks parked overnight in the Rabatak area.
"We put explosives on a fuel tanker. When it exploded, we fired on the trucks," Taliban spokesman Zabiullah Mujahid said by telephone.
Sidiq Azizi, a spokesman for the province, said that many tankers and semitrailers caught fire after the bomb went off around 2 a.m.
Hours later, plumes of thick black smoke still billowed from the area, and firefighters were spraying water on the burning vehicles.
"There was a big boom," Azizi said. "It's possible that it was a magnetic bomb from insurgents. We are investigating."
"I counted 20 fuel tankers burning. There is still a very big fire," said Azizi, who was at the scene. "The weather is very hot, and it's hard to get close to the fire."
The tankers in the convoy were transporting fuel south toward the Afghan capital, Kabul, from neighboring Uzbekistan to the north.
Earlier this week, militants destroyed three NATO supply trucks in Sayd Abad district of Wardak province in eastern Afghanistan.
Convoy traffic in the east has increased since Pakistan reopened its border crossings about two weeks ago. Islamabad blocked NATO supply trucks for seven months in retaliation for U.S. air strikes that killed 24 Pakistani soldiers.
In the suicide blast in the east, a man wearing an Afghan army uniform blew himself up at a checkpoint in Logar province, killing three Afghan troops, said provincial spokesman Din Mohammad Darwesh. The Taliban issued a statement claiming responsibility for the attack in Baraki Barak district.
NATO said that two service members were killed in a roadside bombing Wednesday in the east, while another died Tuesday in an insurgent attack in the south.