Electroencephalogram (EEG)-based emotion recognition is an important research area in affective computing and mental health assessment. To address the insufficient modeling of long-term dependencies in EEG signals, this paper proposes an EEG emotion recognition method based on multi-branch convolutional neural networks (CNN) and Transformer (MCT). The proposed method employs a multi-scale CNN to extract local temporal features from EEG signals and constructs a parallel CNN-Transformer architecture to capture both short-term variations and long-term dependencies, thereby enabling temporal feature modeling at different time scales. Furthermore, a dual-branch convolutional structure is utilized to learn both global and local spatial channel features of EEG signals. A convolutional block attention module (CBAM) is then introduced to fuse the spatio-temporal features of EEG signals. Experimental results show that the proposed MCT model achieves a classification accuracy of 83.83% on the Shanghai Jiao Tong University emotion EEG dataset (SEED). On the music emotion EEG dataset (MEEG), it attains accuracies of 90.00% and 92.62% for the arousal and valence dimensions, respectively. On the database for emotion analysis using physiological signals (DEAP), it attains accuracies of 61.30% and 61.04% for the arousal and valence dimensions. The accuracy results on all three datasets outperform those of the best-performing baseline models. These findings indicate that MCT can effectively learn discriminative features associated with emotional states, providing a new perspective for EEG-based emotion recognition research.