The trade group, which represents the five largest recording companies, has settled more than 400 of those cases for around $3,000 each.
The RIAA does not yet know the identities of those it targeted in its latest round of lawsuits but plans to discover them through court-issued subpoenas.
The trade group turned to these "John Doe" lawsuits in January after an appeals court ruled that Internet service providers like Verizon Communicationsto recording-industry investigators.
Also on Monday, the RIAA said it sued 24 individuals by name after discovering their identities through John Doe suits.
Those sued by name declined offers to settle out of court, the RIAA said.
"We will continue to go the extra mile and seek to resolve these cases in a fair and reasonable manner," RIAA President Cary Sherman said. RIAA members include Warner Music Group, Bertelsmann's BMG, EMI Group, Sony Music and Vivendi Universal's Universal Music Group.