We consider linear precoding in the downlink of a multiuser multiple-input-single-output (MISO) overlay cognitive radio (CR) network, wherein a primary base station (PBS) and a cognitive base station (CBS) share the same frequency band to transmit to a number of primary users (PUs) and secondary users. Conventionally, linear precoding techniques in a CR network aim to limit or completely cancel interference to the PUs while achieving maximum downlink throughput of the CR system. In this paper, we investigate novel adaptive linear precoding techniques at the CBS to exploit the interference to the primary and secondary systems, instead of its cancellation, by exploring the potential of making use of interference energy when the interference between the two systems is mutually constructive on an instantaneous basis. By doing so, the received signal-to-noise ratio (SNR) is enhanced by the signal energy gleaned from the constructive interference. In this direction, constructive-interference-based adaptive linear precoding techniques are proposed for the CR network, assuming a simple zero-forcing precoder at the PBS and noncausal knowledge of the primary message at the CBS. The presented analysis and simulations show that the proposed precoding techniques outperform conventional techniques, as the CR transmission actively enhances the performance of the PUs through constructive interference while achieving significant throughput for its own transmission.