Spiking neural networks (SNNs) offer an energy-efficient alternative to conventional deep learning by mimicking the event-driven processing of the brain. Incorporating the Transformers with SNNs has shown promise for accuracy, yet it is incompetent to capture high-frequency patterns like moving edge and pixel-level brightness changes due to their reliance on global self-attention operations. Porting frequency representations in SNN is challenging yet crucial for event-driven vision. To address this issue, we propose the Spiking Wavelet Transformer (SWformer), an attention-free architecture that effectively learns comprehensive spatial-frequency features in a spike-driven manner by leveraging the sparse wavelet transform. The critical component is a Frequency-aware Spiking Token Mixer (FSTM) with three branches: 1) spiking wavelet learner for spatial-frequency domain learning, 2) convolution-based learner for spatial feature extraction, and 3) spiking pointwise convolution for cross-channel information aggregation. We also adopt negative spike dynamics to strengthen the frequency representation further. This enables the SWformer to outperform vanilla spiking Transformers in capturing high-frequency visual components, as evidenced by our empirical results. Experiments on both static and neuromorphic datasets demonstrate SWformer's effectiveness in capturing spatial-frequency patterns in a multiplication-free, event-driven fashion, outperforming state-of-the-art SNNs. SWformer achieves an over 50 reduction in energy consumption, a 21.1 reduction in parameter count, and a 2.40 performance improvement on the ImageNet dataset compared to vanilla Spiking Transformers.