Identifying new sensing modalities for indoor localization is an interest of research. This paper studies powerline-induced alternating magnetic field (AMF) that fills the indoor space for the orientation-aware three-dimensional (3D) simultaneous localization and mapping (SLAM). While an existing study has adopted a uniaxial AMF sensor for SLAM in a plane surface, the design falls short of addressing the vector field nature of AMF and is therefore susceptible to sensor orientation variations. Moreover, although the higher spatial variability of AMF in comparison with indoor geomagnetism promotes location sensing resolution, extra SLAM algorithm designs are needed to achieve robustness to trajectory deviations from the constructed map. To address the above issues, we design a new triaxial AMF sensor and a new SLAM algorithm that constructs a 3D AMF intensity map regularized and augmented by a Gaussian process. The triaxial sensor's orientation estimation is free of the error accumulation problem faced by inertial sensing. From extensive evaluation in eight indoor environments, our AMF-based 3D SLAM achieves sub-1m to 3m median localization errors in spaces of up to 500 m2, sub-2° mean error in orientation sensing, and outperforms the SLAM systems based on Wi-Fi, geomagnetism, and uniaxial AMF by more than 30%.