Germany's Marcel Siem eagled the 16th hole and took the second-round lead at 10-under 132. India's Jeev Milkha Singh, who had a 70 after opening with a 63, and Denmark's Soeren Hansen (66) were one shot back. Spain's Ignacio Garrido was two strokes behind.
But McIlroy made three straight birdies from Nos. 14 to 16 to finish at 68. McDowell birdied his first two holes in a 66. Both were at 4-under 138, with Clarke at 143.
"If I get off to a good start tomorrow and post a number, then I feel I can give myself a good chance," McIlroy said.
McIlroy has been feuding with Jay Townsend, a U.S. golf commentator and former European pro. On Thursday, Townsend criticized the 22-year-old star and his caddie, J.P. Fitzgerald. After Townsend called McIlroy's opening-day tactics "silly" and Fitzgerald's management "shocking," McIlroy sent him a message on Twitter. He called Townsend a "failed golfer" who should "shut up."
McIlroy said Friday he spoke with his parents and manager about staying calm, but he stuck by his harsh words. Townsend has been providing live fairway commentary for Irish broadcasters RTE.
"I don't really have any respect for the man," McIlroy said. "I have to stand up for J.P. because he's the best man that I think I can have on my bag. He's taken me from 200th in the world to major champion and now fourth in the world."
When asked if he agreed with some commentators' views that he was behaving like a spoiled brat, McIlroy said: "Well, I'm an only child. If that means I'm spoiled, I don't know."
Townsend said Friday he was rebuffed when he tried to talk to McIlroy through his manager.
Clarke dropped four shots in the last eight holes. His final putt for par on the 18th stopped barely an inch short.
"I was trying, trying, trying and couldn't get anything going. I couldn't buy a putt," he said.
The 42-year-old Clarke arrived in Killarney with a hacking cough and cold that he attributed to "self-inflicted man flu." But he said he didn't regret a single pint or all-night party since his breakthrough victory in Sandwich.
"I'm not physically tired, no. Just mentally tired," he said.
Also missing the cut was Ireland's Padraig Harrington, who finished 3 over and appeared distressed as putt after putt rolled narrowly left or right.
Harrington won three majors in 2008 and 2009 but has struggled since. He said his double bogey and bogey on Nos. 6 and 7 "knocked the stuffing out of me."
"It was a good day on the greens," he said. "They didn't drop, but they'll drop someday."