The rapid development of single-modal pre-training has prompted researchers to pay more attention to cross-modal pre-training methods. In this paper, we propose a unified-modal speech-unit-text pre-training model, SpeechUT, to connect the representations of a speech encoder and a text decoder with a shared unit encoder. Leveraging hidden-unit as an interface to align speech and text, we can decompose the speech-to-text model into a speech-to-unit model and a unit-to-text model, which can be jointly pre-trained with unpaired speech and text data respectively. Our proposed SpeechUT is fine-tuned and evaluated on automatic speech recognition (ASR) and speech translation (ST) tasks. Experimental results show that SpeechUT gets substantial improvements over strong baselines, and achieves state-of-the-art performance on both the LibriSpeech ASR and MuST-C ST tasks. To better understand the proposed SpeechUT, detailed analyses are conducted. The code and pre-trained models are available at https://aka.ms/SpeechUT.