Wireless implantable devices capable of monitoring the electrical activity of the brain are becoming an important tool for understanding, and potentially treating, mental diseases such as epilepsy and depression. Compressive sensing (CS) is emerging as a promising approach to directly acquire compressed signals, allowing to reduce the power consumption associated with data transmission. To this end, we propose an efficient CS scheme which exploits the structure of the intracranial EEG signals, both in sampling and recovery. Our structure-aware approach is conceptually simple to implement in hardware and yields state-of-the-art compression rates up to 32x with high reconstruction quality, as illustrated on two human iEEG datasets.