Police said their bodies had been sent to their respective villages in different districts.
The groups are trying to introduce Islamic law in Bangladesh, a predominantly Muslim democracy.
The six were sentenced to death by the High Court in May last year for masterminding or involvement in a series of bomb attacks which killed at least 30 people and wounded 150 in 2005.
The victims included judges, lawyers, police and other officials.
Police said security had been increased across the country as intelligence reports said followers of the condemned men were trying to stage protests mainly in their home districts.
Bangladesh has been under a state of emergency since January 11, and is run by an interim government until the next election - for which no date has yet been set.
The interim government leaders earlier said ahead of the election they would cleanse the country of everything including corrupt politicians and terrorists.