Predicting natural and diverse 3D hand gestures from the upper body dynamics is a practical yet challenging task in virtual avatar creation. Previous works usually overlook the asymmetric motions between two hands and generate two hands in a holistic manner, leading to unnatural results. In this work, we introduce a novel bilateral hand disentanglement based two-stage 3D hand generation method to achieve natural and diverse 3D hand prediction from body dynamics. In the first stage, we intend to generate natural hand gestures by two hand-disentanglement branches. Considering the asymmetric gestures and motions of two hands, we introduce a Spatial-Residual Memory (SRM) module to model spatial interaction between the body and each hand by residual learning. To enhance the coordination of two hand motions wrt. body dynamics holistically, we then present a Temporal-Motion Memory (TMM) module. TMM can effectively model the temporal association between body dynamics and two hand motions. The second stage is built upon the insight that 3D hand predictions should be non-deterministic given the sequential body postures. Thus, we further diversify our 3D hand predictions based on the initial output from the stage one. Concretely, we propose a Prototypical-Memory Sampling Strategy (PSS) to generate the non-deterministic hand gestures by gradient-based Markov Chain Monte Carlo (MCMC) sampling. Extensive experiments demonstrate that our method outperforms the state-of-the-art models on the B2H dataset and our newly collected TED Hands dataset. The dataset and code are available at: https://github.com/XingqunQi-lab/Diverse-3D-Hand-Gesture-Prediction.