The trip through Carroll Gardens/Boerum Hill is 5 miles of bike lanes in the door zone. You have to be really vigilant.

I found this to be the fastest, most direct route.
From the Grand Army Plaza entrance to Prospect Park, follow the roundabout to get yourself to Vanderbilt. (It's slightly hairy getting to Vanderbilt from the park; it's safest to stay in the outside lane of the traffic circle.) Take Vanderbilt (recently given a bike lane and traffic-calming makeover) down across Atlantic all the way to DeKalb. Make a left on DeKalb, which has a bike lane. Make a right on Ashland, also a bike lane. Take it past the BQE overpass and make a left onto Sands. Resist the urge of other cyclists to ride against traffic when the bike lane momentarily veers off to the right and away from the bridge entrance: this "shortcut" saves no time and can really land you in heavy traffic with little wiggle room.