Speech emotion recognition (SER) plays an important role in human-computer interaction, which can provide better interactivity to enhance user experiences. Existing approaches tend to directly apply deep learning networks to distinguish emotions. Among them, the convolutional neural network (CNN) is the most commonly used method to learn emotional representations from spectrograms. However, CNN does not explicitly model features' associations in the spectral-, temporal-, and channel-wise axes or their relative relevance, which will limit the representation learning. In this article, we propose a deep spectro-temporal-channel network (DSTCNet) to improve the representational ability for speech emotion. The proposed DSTCNet integrates several spectro-temporal-channel (STC) attention modules into a general CNN. Specifically, we propose the STC module that infers a 3-D attention map along the dimensions of time, frequency, and channel. The STC attention can focus more on the regions of crucial time frames, frequency ranges, and feature channels. Finally, experiments were conducted on the Berlin emotional database (EmoDB) and interactive emotional dyadic motion capture (IEMOCAP) databases. The results reveal that our DSTCNet can outperform the traditional CNN-based and several state-of-the-art methods.