Seven people were later arrested in connection with some of the bombings.
Police said that more than 200 small bombs exploded almost simultaneously across the country on Wednesday.
Rabiul Alam, 40, a rickshaw puller, was seriously injured when a bomb went off next to him as he waited for a fare outside a college in northwestern Chapainawabganj town. He died on the way to a hospital, a doctor at Rajshahi Medical College Hospital said.
A ten-year-old boy also died when he picked one of the devices at Savar in central Bangladesh, said local police chief Tariq Kamal.
Police said the bombs were homemade and apparently designed to cause only limited damage. But the blasts caused panic and massive traffic jams in a number of cities, as people fled for safety and rushed to schools to bring their children home.
No claims of responsbility
There was no immediate claim of responsibility, but leaflets from a banned Islamic group, the Jumatul Mujahedin, were found at the scenes of a number of explosions, police said. The group wants to establish an Islamic state in Bangladesh, an overwhelmingly Muslim nation governed by secular laws.
"It's an organised attack. It's no simple incident"
Home Ministry official
"It is an organised attack," Lufuzzaman Babar, a top official in the Home Ministry, told the local TV station ATN Bangla. "It is not a simple incident."
Police made a number of quick arrests. Three men were arrested in the southern district of Cox's Bazaar for carrying or hurling bombs, said police officer Rezaul Karim.
In the city of Chittagong, police arrested two men carrying crude homemade bombs and firecrackers, said police officer Osman Gani said. Two other suspects were picked up elsewhere in Chittagong, Gani said without elaborating.