Semantic communications have been utilized to execute numerous intelligent tasks by transmitting task-related semantic information instead of bits. In this article, we propose a semantic-aware speech-to-text transmission system for the single-user multiple-input multiple-output (MIMO) and multi-user MIMO communication scenarios, named SAC-ST. Particularly, a semantic communication system to serve the speech-to-text task at the receiver is first designed, which compresses the semantic information and generates the low-dimensional semantic features by leveraging the transformer module. In addition, a novel semantic-aware network is proposed to facilitate transmission with high semantic fidelity by identifying the critical semantic information and guaranteeing its accurate recovery. Furthermore, we extend the SAC-ST with a neural network-enabled channel estimation network to mitigate the dependence on accurate channel state information and validate the feasibility of SAC-ST in practical communication environments. Simulation results will show that the proposed SAC-ST outperforms the communication framework without the semantic-aware network for speech-to-text transmission over the MIMO channels in terms of the speech-to-text metrics, especially in the low signal-to-noise regime. Moreover, the SAC-ST with the developed channel estimation network is comparable to the SAC-ST with perfect channel state information.