The co-hosts reached a milestone on Sunday (7 February) as the draw for the qualifying competition was held at Warsaw's Palace of Culture and Science, a symbol of old Cold War divisions.
"There is a long list of things that need to be improved, but we are strictly dealing with minor problems," said Platini at a press conference.
"Enormous efforts have been undertaken by Poland and Ukraine and their football federations," he added.
"We have got two years to go until the finals, but to be frank we are not dealing with major problems at the moment. Everything is as it should be," the former French footballer said.
UEFA has generally been happy with Poland's readiness to host the tournament, but problems with stadium construction, infrastructure, communications and accommodation previously prompted Platini to warn Ukraine that it risked losing the right to stage the event (EurActiv 30/07/09).
The tournament is due to take place in June and July 2012 in four cities in each country: Warsaw, Gdańsk, Poznań and Wrocław will stage matches in Poland, with Kiev, Lviv, Kharkiv and Donetsk hosting games in Ukraine.
"It's three years since UEFA awarded the hosting of the 2012 European football championships to Poland and Ukraine," Polish Prime Minister Donald Tusk said. "These three years have been a period of hard work in both countries," he added.
"Euro 2012 infrastructure is being constructed, including stadia, airports, motorways and hotels," Tusk, an ardent football fan, insisted.
Among the issues still to be resolved, Platini said, is which Ukrainian cities will stage quarter-finals, before adding: "We have two years to decide that, it is not a major worry."
Euro 2012 will be the first time that UEFA's flagship tournament has taken place behind the former Iron Curtain.
(EurActiv with Reuters.)