The call for demonstrations by the Indian arm of the group follows a March 29 court order issued in the southern city of Chennai demanding 15 Indian Internet providers block access to file-sharing websites such as Pirate Bay.
The order has resulted in access being denied to a host of websites that carry pirated films and music among other legal content, including www.isohunt.com and www.pastebin.com.
On Wednesday, the Anonymous forum fired an opening shot by attacking the website of state-run telecom provider MTNL, pasting the logo of the group -- the mask of 17th century revolutionary Guy Fawkes -- on www.mtnl.net.in.
In an open letter the same day, the group accused the government of trying to create a "Great Indian Firewall" to establish control on the web and issuing a "declaration of war from yourself... to us."
Internet users and supporters have been asked to join peaceful rallies in cities including the capital New Delhi and the tech hub of Bangalore.
Tech website www.pluggd.in reported the demonstrators have been asked to wear Guy Fawkes' masks, download a recorded message to play to police, and are to chant "United as one! Divided as zero! We are Anonymous! We are legion!"
The new rules regulating Internet companies -- providers, websites and search engines -- instruct them that they must remove "disparaging" or "blasphemous" content within 36 hours if they receive a complaint by an "affected person".
Anonymous is a secretive 'hacker-activist network and is thought to be a loosely knit collective with no clearly defined leadership structure.