Just saw this article and thought that I'd pass it along. http://news.yahoo.com/chicagos-beer-gardens-nations-best-lists-185000838.html And soliciting your thoughts.
Fountainhead is definitely number one in my opinion... Sheffields is great, and The Daily is a really nice little garden they just could use an upgrade on their beer list.
Cork and Kerry has a really nice one. Not a big craft selection but usually a couple good brews. I also like the beer garden at Duke of Perth, again not a huge craft selection but the Bellhaven options are good and a great selection of Scotch.
Quenchers has a nice little outdoor space, but you're still very close to the sometimes noisy Western Ave. Moody's Pub is almost bucolic, but I think the only craft beers they have are Anchor Steam and 312.
Agree with everything else above. Some additional ones to check out (very Logan Square-ish heavy FWIW): Kuma's Boiler Room - solar panels are set up, and just inside the doors to the patio there is a screen with energy usage Chief O'Neils - beer garden is ginormous. Beer selection is decent Ada Street - out door ping pong table Longman & Eagle / Off Site Bar Parson's Chicken & Fish (not yet open) - another venture by the L&E team... the outdoor space looks great though, supposedly for about 150 people. Small Bar Logan Square Village Tap