EEG-based recognition of activities and states involves the use of prior neuroscience knowledge to generate quantitative EEG features, which may limit BCI performance. Although neural network-based methods can effectively extract features, they often encounter issues such as poor generalization across datasets, high predicting volatility, and low model interpretability. Hence, we propose a novel lightweight multi-dimensional attention network, called LMDA-Net. By incorporating two novel attention modules designed specifically for EEG signals, the channel attention module and the depth attention module, LMDA-Net can effectively integrate features from multiple dimensions, resulting in improved classification performance across various BCI tasks. LMDA-Net was evaluated on four high-impact public datasets, including motor imagery (MI) and P300-Speller paradigms, and was compared with other representative models. The experimental results demonstrate that LMDA-Net outperforms other representative methods in terms of classification accuracy and predicting volatility, achieving the highest accuracy in all datasets within 300 training epochs. Ablation experiments further confirm the effectiveness of the channel attention module and the depth attention module. To facilitate an in-depth understanding of the features extracted by LMDA-Net, we propose class-specific neural network feature interpretability algorithms that are suitable for event-related potentials (ERPs) and event-related desynchronization/synchronization (ERD/ERS). By mapping the output of the specific layer of LMDA-Net to the time or spatial domain through class activation maps, the resulting feature visualizations can provide interpretable analysis and establish connections with EEG time-spatial analysis in neuroscience. In summary, LMDA-Net shows great potential as a general online decoding model for various EEG tasks.