We propose a novel power saving strategy called Sleep during Neighbor-Addressed Frame (SNAF) for improving energy efficiency of IEEE 802.11 based wireless networks. IEEE 802.11 (Wi-Fi) radios that employ SNAF mode can turn OFF their wireless transceivers (i.e., put radios in sleep mode) within specific periods of neighbor-addressed frames while they are being received. The sleep duration of transceivers is easy to determine with no loss of packet's critical control information. The proposed SNAF mode operation does not have any negative effect on network throughput and even complements Power Saving Mode (PSM) available in 802.11 standard. We further propose GreenFrame format for next generation wireless networks. In experiments conducted in wireless LAN scenarios, we observed savings as much as 57.8% when we implement SNAF mode in 802.11 standard and up to 49.5% when we implement SNAF mode in 802.11 PSM. ©2010 IEEE.