My recommendations for utmost efficiency...
As per CSARdiver, bribe the cops as often as possible, and only start shooting when absolutely necessary to escape or enter a bank. Drop off your ill-gotten goods as little as possible to shed minutes off. Remember that you don't get credit for the heist until you hit your safehouse, so be cognizant of your heat and health. Also, faster vehicles are, as always, your friend, so avoid pickups if possible.
You will spawn initially to the west of the northernmost bank in Midtown. Bomb it, steal the cash, and immediately make your way to the southern Midtown bank via Fourth. Rinse and repeat, then make your way to the northernmost bank in Little Italy, near your initial safehouse.
Travel down Fourth, before making your way onto Bowery, then turn onto Grant. Bribe, rob, and then, as you're close to your safehouse, drop off and top off your health if necessary.
From here, use the Manhattan Bridge on Canal Street to get to and from Brooklyn's bank. I'd recommend turning right onto Tillary right off the bridge, hitting a right on Main, then another right onto Washington, so your car is facing the proper getaway.
After Brooklyn, dip south on Broadway to hit the Little Italy bank on Broadway and Worth, then jam back north on Broadway to westbound Canal Street, and put the pedal to the metal to get across to Jersey. Blast through the Holland Tunnel, keep straight on Washington, turn left on 4th and then a right onto Park. Bribe, rob, and jet a quick block north to drop off your last three banks worth of cash, and viola, achievement unlocked.
Took me just over fifteen minutes, mainly because I didn't heed CSARdiver's, or my own, advice and went in guns-blazing once, forcing me to ditch my flaming sportscar when the cops tried to take me out, and had to jump into a larger, slower sedan for a bit.