One man died in Chandpur, 170km southeast of the capital, Dhaka, police superintendent B M Harunur Rashid said by telephone.
He said three people, including a lawyer, were wounded in Monday's blast, and that two people suspected of carrying the bomb were detained.
Also on Monday, another man was killed and six were wounded in a bomb blast at a court building in nearby Laxmipur town, police said.
And one or more bombs went off at a court in the port city of Chittagong, police added. There was no immediate word on casualties there.
Police said they were checking a report of bomb blasts in one or two other places.
About 500 small bombs exploded across Bangladesh on 17 August, killing two people and wounding nearly 100.
The authorities blamed the attack on Jamaat-ul-Mujahidin, an Islamist group they banned last February, along with another group, Jagrata Muslim Janata.
Local reporters said the suspects detained after Monday's blasts in Chandpur and Laxmipur said they were members of Jamaat-ul-Mujahidin.
Police said they were interrogating them.