The use of electrodynamic tethers for propulsion and power generation is attractive for missions to the outer planets, which are traditionally handicapped by large propellant requirements, large times of flight, and a scarcity of power available. In this work, the orbital dynamics of a spacecraft using electrodynamic tether propulsion during the mission phases of capture, apojove pump down and perijove pump up, in the Jovian system are investigated. The main result is the mapped design space involving mission duration, tether length, and minimum perijove radius. Phase-free flyby sequences are also included, which provide performance upper bounds for a given mission architecture. It is found to be advantageous to use inbound-only flybys of the Galilean moons during capture. Flybys during the apojove pump down phase are only useful in conjunction with a perijove-raising mechanism at apoapse such as solar perturbations or a small propulsive maneuver. The electrodynamic tether system is also shown to be capable of lowering the spacecraft's orbit to a Europa-Ganymede Hohmann orbit with a total flight time of under a year and a half after entering Jupiter's sphere of influence, using reasonable assumptions for spacecraft mass and tether parameters.