The team filled the roster openings by promoting safety Vince Anderson, receiver Dezmon Briscoe and guard Brandon Carter to the active roster from the practice squad on Tuesday.
Grimm broke his left leg during last Sunday's 17-10 loss at Baltimore. Joseph broke his left foot against the Ravens.
The opening created by Piscitelli's release was filled by signing safety Larry Asante from Cleveland's practice squad.
The Bucs also added two players to their practice squad, receiver Ed Gant and defensive end E.J. Wilson.