The neural transducer is an end-to-end model for automatic speech recognition (ASR). While the model is well-suited for streaming ASR, the training process remains challenging. During training, the memory requirements may quickly exceed the capacity of state-of-the-art GPUs, limiting batch size and sequence lengths. In this work, we analyze the time and space complexity of a typical transducer training setup. We propose a memory-efficient training method that computes the transducer loss and gradients sample by sample. We present optimizations to increase the efficiency and parallelism of the sample-wise method. In a set of thorough benchmarks, we show that our sample-wise method significantly reduces memory usage, and performs at competitive speed when compared to the default batched computation. As a highlight, we manage to compute the transducer loss and gradients for a batch size of 1024, and audio length of 40 seconds, using only 6 GB of memory.