The blasts occurred in Bujumbura's central business district, and reports suggest at least four people were injured.
Burundi has been in crisis since April 2015 when President Pierre Nkurunziza ran for a controversial third term.
This led to ongoing violence and fears that Burundi is sliding into ethnic conflict.
The BBC's Prime Ndikumagenge, in the capital, says the latest attacks are unusual because they occurred in daylight, and in the centre of the city, whereas most violence happens at night.
Two grenades were hurled around midday, near the central post office and a building hosting a mobile phone operator, injuring four people.
The third grenade was detonated at a bus station south of the capital, behind what used to be the central market, our correspondent says.
At least 439 people have died and 240,000 have fled abroad since last April, the UN says.
Last month, human rights group Amnesty International published satellite images it said were believed to be five mass graves near Burundi's capital, where security forces were accused of killing scores of people in December.
A fact-finding mission by the AU has reported arbitrary killings, torture and the "closure of some civil society organisations and the media".
Burundi on the brink
Tit-for-tat killings spread fear
Burundi's football-playing president
Presidents who cling to power
10.4m population
50 years - life expectancy for a man
2nd poorest country in the world
85% are Hutu, 14% Tutsi
300,000 died in civil war
Find out more about Burundi
Three grenade explosions have rocked the capital of the troubled central African nation of Burundi.