The trio sit on nine-under par, with Cameron Percy and Ryan Palmer a shot back, and group of five players that includes Nick Watney and Hunter Mahan two adrift on seven-under.
Overnight rain left the TPC Four Seasons course heavily saturated and turned the 406-yard par-four 14th into a 104-yard par-three after large patches of the fairway unplayable.
Walker's round of 66 began slowly with a birdie on the seventh the lone deviation from par on the opening nine before producing an up and down inward nine.
The 36-year-old, who is ranked 12th in the world, made birdie on the 11th before recording his first bogey of the round on the next hole.
He bounced back with successive birdies before registering his second bogey on the 15th, but a birdie on the 16th ensured he ended on a high.
Overnight leader Bowditch began his second-round 68 with consecutive pars before recording two bogeys. The Australian recovered well by sinking a hat-trick of birdies from the fifth to the seventh hole to round off his front nine.
Like Walker, Bowditch failed to remain consistent on the back nine as he recorded four birdies and four bogeys from the 10th to 17th.
Curran meanwhile started his second-round 63 with a bogey on the first before rallying back with three pars and a birdie over his next four holes. However, he made his second bogey of the round on the sixth before launching another comeback with a birdie on the seventh a hat-trick of birdies from the ninth to the 11th hole.
Curran maintained his red-hot form throughout the remainder of the back nine as he made his second hat-trick of birdies from the 14th to the 16th hole before finishing with pars on the 17th and 18th.
Sitting one stroke behind the trio are Australia's Cameron Percy and Ryan Palmer, who hit second rounds of 64 and 66 respectively.
Meanwhile, Spieth rocketed up the leaderboard as he followed up his first-round 69 with a 64.
Spieth began on a high with two birdies on his first three holes before registering a bogey on the sixth. He atoned for his mistake with three consecutive birdies from the seventh to the ninth hole and went on to finish with two more birdies and a bogey on his back nine.
Defending champion Brendon Todd missed the cut after finishing tied 99th following a second-round 68, which consisted of four birdies and three bogeys.