In this paper, we present a pose estimation strategy for the end effector of a tensegrity manipulator, based on the use of an extended Kalman filter and a deep feedforward neural network with three hidden layers. Our scheme is based on the fusion of sensor data obtained from an inertial measurement unit and ArUco fiducial markers. The method was implemented on a six bar tensegrity prism manipulator, tested using ground truth acquired from an external vision-based motion capture system, and compared with other estimation methods. The experimental results show the ability of our method to provide reliable pose estimates, also dealing with the problems caused by the tensegrity structure, including marker occlusions due to the presence of bars and strings.