The motorcycle was detonated by remote control at a busy market in Janikhel district in Paktika province, said Mohammad Rahman Ayaas, spokesman for the provincial governor. He said a woman was among the three killed in the explosion, which also wounded 19 people.
No group immediately claimed responsibility for the bombing but he blamed the Taliban, AP reported.
The insurgents frequently use suicide attacks and roadside bombs to target convoys of Afghan security forces across the country. Civilians are often victims of such explosions.
The Taliban ambush that killed four people took place in Alasay district in northeastern Kapisa province, said Massoud Chardara, deputy chief of police. The Taliban had ambushed a military convoy and in subsequent clashes a Taliban mortar hit a house, killing four people inside it.
The Taliban sent a text message to the media, blaming Afghan government forces for the civilian deaths.
Separately, the Ministry of Interior says Afghan security forces have killed 12 militants of the ISIL group, including a key commander, in eastern Nangarhar province in the past 24 hours.