Deep learning systems have been applied mostly to Euclidean data such as images, video, and audio. In many applications, however, information and their relationships are better expressed with graphs. Graph Convolutional Networks (GCNs) appear to be a promising approach to efficiently learn from graph data structures, having shown advantages in many critical applications. As with other deep learning modalities, hardware acceleration is critical. The challenge is that real-world graphs are often extremely large and unbalanced; this poses significant performance demands and design challenges. In this paper, we propose Autotuning-Workload-Balancing GCN (AWB-GCN) to accelerate GCN inference. To address the issue of workload imbalance in processing real-world graphs, three hardware-based autotuning techniques are proposed: dynamic distribution smoothing, remote switching, and row remapping. In particular, AWB-GCN continuously monitors the sparse graph pattern, dynamically adjusts the workload distribution among a large number of processing elements (up to 4K PEs), and, after converging, reuses the ideal configuration. Evaluations are performed using an Intel D5005 FPGA with five commonly-used datasets. Results show that 4K-PE AWB-GCN can significantly elevate the average PE utilization (from 32.5% to 88.6%) and demonstrate considerable performance speedups over CPUs (7569x), GPUs (80.3x), and a prior GCN accelerator (7.4x).