According to Eikenberry, giving circles aren't a huge phenomenon -- about 400 of them have been identified nationwide -- but they're nevertheless an interesting and growing trend. The idea is that people band together to make pooled, group gifts to an organization. The gifts most typically are one-time deals, but some circles have gotten interested enough to provide ongoing volunteer help or other support.
Eikenberry does a nice job of laying out the advantages and disadvantages of receiving funds from a giving circle. The advantages include a potential new source of not just funding but interested (and well-connected) donors and volunteers, and added prestige at being chosen. And your organization may not have to do much work to get the gift, much less comply with followup reporting requirements!
The disadvantages seem largely to do with the fact that these circles are made up of human beings, with all their various frailties and personalities, and few of them knowing much about how nonprofits work. Some of the nonprofit staff interviewed by Eikenberry reported frustration at spending a lot of time responding to a giving circle's requests for information and such, only to have the circle lose interest or shift focus before making a gift.
The bottom line? You probably shouldn't shift fundraising focus to currying favor from giving circles -- they, in most cases, want to find you, not the reverse. Just keep doing what you're doing, try to maintain a high community profile, and if you're lucky, a giving circle -- one of the more professional ones, with good follow-through -- will find you.