Nishikori won just 14 points in the first set and nine in the second, losing to Thiem for the first time.
Alexander Zverev has fallen in love with the Yorkshire accent after winning a gruelling second-round encounter with Dusan Lajovic at the French Open.
Dzumhur, the world number 29, was in the middle of a tough five-set match at the French Open, against second seed Alexander Zverev.
The 24-year-old Thiem leads 4-2 overall against Zverev, but lost to him last month in the Madrid Masters final.
"Well, I'm young, so I might as well stay on court, get some practice and entertain you guys", smiled Zverev, who is bidding to become the first German man to win the French Open since 1937. "All the hours in the gym have paid off".
Zverev, 21, who is seen as the main threat to Rafael Nadal´s expected coronation as champion for an 11th time, endured a nightmare outing on Court Philippe Chatrier that lasted nearly four hours.
He will face French No.15 seed Lucas Pouille or Karen Khachanov in the fourth round.
Dzumhur, seeded 26th, had a match point on Zverev's serve at 5-4 in the final set.
"I think that's the matchup most of the fans in Germany and Austria were hoping for when they saw the draw", he said.
Djokovic, seeded 20 after a year of struggles with an elbow injury and mediocre form, was never in trouble against an erratic Verdasco, who needed treatment for a badly blistered left foot at the end of the second set. Djokovic advanced to his 40th Grand Slam quarter-final. "I'm very happy about being in the quarterfinals here, going the hard way, going the long distance ... showing everybody that I can play for as long as I need to".
On the men's side, Novak Djokovic was facing Fernando Verdasco on Court Philippe Chatrier, where he won the last of his 12 major titles in 2016. But Dzumhur felt it was within his reach and also ran out to catch it.
Stephens next plays either Australian Open champion Caroline Wozniacki - seeded second - or 14th-seeded Russian Daria Kasatkina.
The third-seeded pairing of Andreja Klepac and Maria Jose Martinez Sanchez beat the Americans 6-4, 6-7 (4), 6-0 to advance to the quarter-finals.