DERRY, N.H. — Gov. John R. Kasich of Ohio unabashedly promotes his expansion of Medicaid under Obamacare, shows little appetite for relitigating culture-war battles like same-sex marriage and offers not much more than a shrug when asked about Hillary Rodham Clinton’s turning over her email server to the F.B.I.

“I’m really more concerned about letting people know who I am, rather than that much about Hillary,” he said after a town hall-style meeting Tuesday night, the political equivalent of letting a batting practice strike go by without a swing.

At a moment when many of his Republican rivals are vying to criticize President Obama and Mrs. Clinton most sharply and to position themselves as the most ideologically pure candidates in the field, Mr. Kasich is taking the opposite approach. And it is paying off — at least in this quirky but crucial state where answers to questions about transportation funding can become applause lines, “Lake Wobegon” references get knowing laughs and Republicans can grow animated about their disdain for negative advertising.

Just a month after entering the race, Mr. Kasich is rising in the polls in New Hampshire, winning head-turning endorsements and drawing new voters to his events who were impressed with his debate performance last week. And his gains could have a significant ripple effect on the Republican primary in this state and beyond, because he is appealing to many of the same voters that former Gov. Jeb Bush of Florida and Gov. Chris Christie of New Jersey are targeting.