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In multi-tiered fog computing systems, to accelerate the processing of computation-intensive tasks for real-time IoT applications, resource-limited IoT devices can offload part of their workloads to nearby fog nodes, whereafter such workloads may be offloaded to upper-tier fog nodes with greater computation capacities. Such hierarchical offloading, though promising to shorten processing latencies, may also induce excessive power consumptions and latencies for wireless transmissions. With the temporal variation of various system dynamics, such a trade-off makes it rather challenging to conduct effective and online offloading decision making. Meanwhile, the fundamental benefits of predictive offloading to fog computing systems still remain unexplored. In this paper, we focus on the problem of dynamic offloading and resource allocation with traffic prediction in multi-tiered fog computing systems. By formulating the problem as a stochastic network optimization problem, we aim to minimize the time-average power consumptions with stability guarantee for all queues in the system. We exploit unique problem structures and propose PORA, an efficient and distributed predictive offloading and resource allocation scheme for multi-tiered fog computing systems. Our theoretical analysis and simulation results show that PORA incurs near-optimal power consumptions with queue stability guarantee. Furthermore, PORA requires only mild-value of predictive information to achieve a notable latency reduction, even with prediction errors.