The MPs began receiving an a avalanche of emails after newspaper Expressen published a protest form on its website for readers to fill in.
Once completed, the forms are automatically forwarded to all parliamentarians who supported the new law or failed to turn up in parliament for the vote.
Former Justice Minister Thomas Bodström told Expressen that there was "more action in the parliament than there has been for ten years".
Political commentator Stig-Björn Ljunggren told the newspaper that the powerful response to the campaign could have "a massive effect".
The new signal surveillance law will enable Sweden's National Defence Radio Establishment (Försvarets Radioanstalt - FRA) to monitor all outgoing and incoming communications crossing Sweden's borders from January 1st next year.