Null-steering transmit beamformers aim to maximize the received signal power at the direction of the access point (AP) while inflicting minimal interfering effect on unintended receivers. However, existing null-steering beamformers may not be directly applied to wireless sensor networks (WSNs) as they require every node to be aware of the locations of other nodes in the network. In this paper, a novel null-steering beamformer is introduced that can be implemented in uniformly distributed WSNs in which each node is oblivious of other nodes' locations. The average beampattern expression of the proposed beamformer is derived and its properties are analytically studied. In particular, it is shown that the average beampattern of the proposed beamformer is similar to that of the conventional beamformer in directions with far angular distance from any unintended receiver. The analytical results are also adopted to design a null-steering beamformer with a single null and required average beampattern properties. The optimal null position is obtained which minimizes the maximum sidelobe of the average beampattern while having negligible deteriorating effect on the received power at the AP. Finally, computer simulations are used to validate the analytical results as well as to confirm the effectiveness of the design approach.