Recently, self-attention-based transformers and conformers have been introduced as alternatives to RNNs for ASR acoustic modeling. Nevertheless, the full-sequence attention mechanism is non-streamable and computationally expensive, thus requiring modifications, such as chunking and caching, for efficient streaming ASR. In this paper, we propose to apply RWKV, a variant of linear attention transformer, to streaming ASR. RWKV combines the superior performance of transformers and the inference efficiency of RNNs, which is well-suited for streaming ASR scenarios where the budget for latency and memory is restricted. Experiments on varying scales (100h - 10000h) demonstrate that RWKV-Transducer and RWKV-Boundary-Aware-Transducer achieve comparable to or even better accuracy compared with chunk conformer transducer, with minimal latency and inference memory cost.