Taoiseach Enda Kenny was one of the world leaders that Mrs May talked to in the hours after her appointment as the UK's new prime minister.
Mr Kenny said he congratulated his counterpart during a phonecall on Wednesday evening.
They talked about their "joint support" of the Northern Ireland peace process.
"We discussed the importance of continuing to work together to build on the excellent relationship that now exists between Ireland and Britain," he added.
Mrs May stressed her "commitment to delivering the will of the British people to leave the EU", a Downing Street spokesperson said.
She explained that time would be needed to "prepare for these negotiations" and said she hoped they could be "conducted in a constructive and positive spirit", the spokesperson added.
Mr Kenny said he "outlined Ireland's key areas of concern" over a future UK withdrawal from the EU and gave his perspective on "the challenges that lie ahead" in negotiations on the issue.
He added that he would meet Mrs May "in the coming weeks".
As well as speaking to Mr Kenny, Mrs May took calls from German Chancellor Angela Merkel and French President Francois Hollande.
She also began shaping her new cabinet, appointing Philip Hammond and Boris Johnson as chancellor and foreign secretary respectively.
Theresa May has agreed to build on the "strong relationship" between her country and the Republic of Ireland, the Irish prime minister has said.