Four years after losing 4-0 to Spain in the final, the Azzurri will face Belgium in Lyon on June 13, before hosting Sweden four days later in Toulouse and Ireland on June 22 in Lille.
Conte saw his side overcome a few ups and downs to secure qualification with a game to spare, but the Italian -- who succeeded Cesare Prandelli after last summer's disastrous World Cup -- already sees trouble ahead.
"Belgium are an excellent team, I think they will be among the favourites for the final stages," said Conte, whose side lost 3-1 to Marc Wilmots' men in Brussels last month.
Italy fans will also remember the last time they came up against Sweden in the tournament.
It was 2004 and striker Zlatan Ibrahimovic hit an 85th minute equaliser to secure a 1-1 draw and rob the Azzurri of a possible three points in their penultimate group game.
Despite a subsequent 2-1 win over Bulgaria, Italy were left third in Group C and on their way home as Sweden and Denmark, who drew their final game 2-2, advanced due to a better head-to-head record.
"We all know about the strength of Zlatan," said Conte.
Paris Saint Germain striker Ibrahimovic was not part of the Sweden side that lost 1-0 to Italy in a November 2009 friendly, when Giorgio Chiellini provided the winner.
Italy won eight of their 13 previous games against the Republic of Ireland, including a 2-0 win in Poznan on their way to the Euro 2012 final against Spain.
In the sides' only meeting since then they played to a scoreless draw at Fulham's Craven Cottage ground in May 2014.
Conte added: "And Ireland are certainly a team who play with a lot of emotion."