Time Series Forecasting (TSF) is an important application across many fields. There is a debate about whether Transformers, despite being good at understanding long sequences, struggle with preserving temporal relationships in time series data. Recent research suggests that simpler linear models might outperform or at least provide competitive performance compared to complex Transformer-based models for TSF tasks. In this paper, we propose a novel data-efficient architecture, GLinear, for multivariate TSF that exploits periodic patterns to provide better accuracy. It also provides better prediction accuracy by using a smaller amount of historical data compared to other state-of-the-art linear predictors. Four different datasets (ETTh1, Electricity, Traffic, and Weather) are used to evaluate the performance of the proposed predictor. A performance comparison with state-of-the-art linear architectures (such as NLinear, DLinear, and RLinear) and transformer-based time series predictor (Autoformer) shows that the GLinear, despite being parametrically efficient, significantly outperforms the existing architectures in most cases of multivariate TSF. We hope that the proposed GLinear opens new fronts of research and development of simpler and more sophisticated architectures for data and computationally efficient time-series analysis.