Magnetic resonance electrical property tomography (MR-EPT) has significant potential for the estimation of the electrical properties (EPs) of tissue, which are essential for the calculation of specific absorption rates (SAR), a critical safety factor requiring monitoring and controlling in applications of ultrahigh field magnetic resonance imaging. In this paper, a novel, efficient method based on integral equations is proposed for the calculation of the EPs and the RF-coil-induced Bz field. An inverse problem framework is first constructed to include the forward problem operator, while the EPs are reconstructed by using a nonlinear conjugate gradient method. The RF-coil-induced Bz component is then calculated based on the achieved EPs and the forward operator. The proposed MR-EPT algorithm improves upon and differs from the existing methods in three aspects. First, a three-dimensional algorithm with improved efficiency is proposed. The higher efficiency arises from using a fast integral equation solver as well as an approximation of initial solution. Second, in addition to the EP values, the proposed method calculates the RF-coil-induced Bz component, which is usually neglected in the existing MR-EPT algorithms. Here, we show that considering this Bz field can significantly improve the accuracy of the SAR calculation. Finally, in contrast to differential approaches, the proposed method is more robust against noisy measurement of the transmit magnetic fields, because of the nature of the integral equations. The proposed method is verified through a full-wave simulation and an anatomically accurate numerical brain model, demonstrating its accuracy and efficiency.