Simulating three-dimensional (3D) crack propagation in solid structures poses significant challenges due to the unpredictability of crack paths, complicating both computation and solution strategies. Traditional methods often face difficulties in accurately capturing arbitrary crack propagation during large deformations. The finite particle method (FPM), based on vector mechanics, offers a novel numerical approach for analyzing complex behaviors in solid mechanics. Different from conventional continuum-based methods, FPM discretizes the solid domain into a collection of finite particles, each governed by Newton's second law of motion. This particle-based formulation enables seamless transitions between continuum and non-continuum behaviors by dynamically adding or removing particles, providing significant advantages for crack propagation analysis in both static and dynamic scenarios. In this study, the FPM is extended to address the dynamic fracture in 3D solids, focusing on the challenges related to crack initiation, propagation, and branching. The FPM is combined with an extrinsic cohesive zone model (CZM) to capture the complex behaviors of fractures, avoiding the need to pre-define crack paths and effectively managing discontinuities caused by crack propagation. A discriminant criterion is developed to identify the onset of crack initiation, and an automated embedding process for cohesive elements is implemented to enable real-time simulation of fracture surfaces. To manage the evolving topologies that arise from crack propagation, we propose a general strategy based on an ergodic search algorithm, which updates the connectivity of the discretized solid model dynamically as cracks evolve. In addition, we develop a GPU-based parallel solver using the CUDA toolkit to significantly accelerate fracture computations. The accuracy and applicability of the proposed method are validated through several numerical examples, including fracture simulations of plates and beams subjected to dynamic loading. The results demonstrate the capability of the method to accurately capture the intricate details of crack initiation, growth, and interaction in 3D solids. This extended FPM framework serves as a robust tool for analyzing dynamic fractures in engineering applications, providing a versatile framework for studying delamination, material failure, and structural collapse in both research and practical settings.