Reporting from Kabul at the scene of the blast, Al Jazeera's correspondent in Afghanistan, James Bays, said Achikzay had been investigating links between the Taliban and Pakistan.
The attack took place during rush hour just before shops opened in the Asmayee Wat area of the city, near government and business buildings.
"In the attack four innocent civilians were martyred and another 12 civilians were wounded. The intelligence director survived the attack," the intelligence service said in a statement.
Security forces sealed off the area and sifted wreckage from the blast.
Sher Mohammed, a witness, said: "I saw a motorcycle park in front of two security pickup trucks and explode.
"I helped carry three bodies. Four people were also wounded."
In other news, four Afghan medical workers were kidnapped from Zhari district, 30km west of Kandahar city, on Tuesday, Afghan health officials said.
Abdul Qayoom Pakhoa, a health department spokesman, said: "The team from Kandahar Health Department was abducted by unknown gunmen."
Violence has risen in Afghanistan with the end of winter as the Taliban step up a campaign of suicide attacks.
Afghanistan last year suffered the worst fighting since US-led multinational forces removed the Taliban from power in 2001.