Some were calling for the execution of the bloggers, whom they accuse of insulting Islam.
Last week, blogger Ahmed Rajib Haider was killed outside his home.
The protests come days after authorities blocked some websites for “hurting religious feelings”.
In the capital, Dhaka, thousands of protesters from an alliance of Islamic parties went on a protest rally soon after Friday prayers in the country’s national mosque.
Police fired rubber bullets and tear gas to disperse protesters who threw stones and vandalised buildings, the BBC‘s Anbarasan Ethirajan reports from Dhaka.
Similar violent clashes were reported across the country. In some places, there were clashes between supporters of the ruling Awami League and Islamist activists, our correspondent reports.
Following Mr Haider’s death last week a series of blog posts described as anti-Islamic have circulated, but their authorship was not immediately clear.
The clashes come amid renewed tension over a tribunal judging war crimes committed during the 1971 independence war.
Last week, Bangladesh’s parliament amended a law which will allow the state to appeal against the life sentence given by the tribunal to an Islamist party leader.
In recent weeks a group of bloggers, including Mr Haider, had launched mass protests demanding his execution and a ban on the country’s largest Islamist party, the Jamaat-e-Islami.
Islamist parties have called for a nationwide general strike on Sunday in protest at the killing of their supporters in recent clashes.
The Islamist party denies being involved in war crimes and says the tribunal is part of a government vendetta against the party.
Human rights groups have said the tribunal falls short of international standards.
Official estimates say more than three million people were killed in the 1971 war which resulted in independence from Pakistan.