Four of the explosions ripped through busy markets in Guwahati while three others went off in the western town of Kokrajhar.
Fourteen people were killed in the first explosion in front of the deputy commissioner's office in Guwahati. Twenty-two others were killed in the second explosion at Ganeshguri near the state secretariat and four in Panbazar in central Guwahati, police say.
They say that 17 people died in the three explosions at Kokrajhar and five more in another explosion at Barpeta Road 130km (81 miles) from Guwahati in the west of the state.
Another blast was reported from near the oil refinery at Bongaigaon, between the towns of Kokrajhar and Barpeta Road.
Most of the bombs were planted in cars, police say.
"A huge blast hit the bus just before the one I was travelling in. The bus got burnt at the front and pedestrians pulled out a lot of people hit by splinters and some suffering burns as well," said engineering student Zinnia Brahma, an eyewitness in Guwahati.
A witness in Kokrajhar, Sanchaita Chakrabarty, told the BBC that a bomb went off just after she had bought vegetables from the local market.
"It happened as I walked away," she said. "The man I had bought the vegetables from was dead, many others around him were seriously injured. Just see how lucky I am to be alive."
Angry crowds attacked the police with stones after the blasts in Guwahati. Dozens of people were injured in the clashes, Reuters reports.
Indian Prime Minister Manmohan Singh, who represents Assam in the upper house of the Indian parliament, is going to the state on Friday.
Intelligence officials blamed Ulfa for the blasts.
But a statement issued by self-styled "lieutenant" Anjan Borthakur on behalf of Ulfa's "Military Council" said that the group was in no way connected with them.
The statement said Ulfa had been blamed as part of a "disinformation campaign" against the separatist group.
Most fighters in one of Ulfa's "strike battalions" have announced a ceasefire with the government - but two other battalions have not laid down their arms and the security forces have attacked and killed many of them in recent weeks.
"So the Ulfa is striking back in a massive way by taking on soft targets," Assam police chief RN Mathur said.
"No other group can trigger so many blasts in so many places in such a co-ordinated fashion."
He said most of the bombs had been planted in cars.
For the past 10 years, Ulfa has been blamed for bombings targeting gas and oil pipelines, oil depots and areas populated by migrant workers.
Assam police intelligence chief Khagen Sharma told the BBC that the latest bombings proved that Ulfa was "desperate for survival and does not mind killing even local people indiscriminately".
Failed peace talks
The group began an armed rebellion against what it describes as colonial rule by Delhi in 1979. Thousands of people have died in the violence.
An effort to start peace talks between the rebels and the Indian government broke down in 2006.
The rebels are seeking a separate homeland for the Assamese people and demanding that non-indigenous people, particularly Hindi and Bengali speakers, leave Assam.
There have been a number of major bomb attacks in India in the past few months, many of them blamed on local Islamist groups.
But local separatists have been held responsible for recent explosions in north-eastern cities.
Two north-eastern state capitals - Agartala in Tripura and Imphal in Manipur - saw serial explosions this month.
At least 20 people were killed and more than 100 injured in the blasts.
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