Group A teams Poland, the Czech Republic, Russia and Greece were already at their base camps, with the co-hosts due to take on Greece in the capital Warsaw and Russia set to play the Czechs in the southwestern city of Wroclaw on Friday.
Group B teams Denmark, Germany, the Netherlands and Portugal were due to arrive in Poland on Monday.
Greece jetted in to the Polish capital Warsaw on Sunday, with the pilot expressing the hope of a nation that is in desperate need of some good news, after months of financial and political turmoil.
"I wish you a good tournament and, when you step onto the pitch, may your eyes see only sky blue and white," UEFA.com reported the flight's captain as saying. "We Greeks need it," he added.
The Greeks were surprise winners of Euro 2004 in Portugal and defender Jose Holebas said that no-one should write them off this time round.
"In football, as in life, you never say never," he was quoted as saying by Real Sports. "Everybody saw what happened this season in the Champions League, when all were predicting a Real Madrid-Barcelona final.
"With team effort and hard work, we can do well."
In Group B, Denmark and the Netherlands will be buoyed by their final warm-up wins. Morten Olsen's Danes beat Australia 2-0 in Copenhagen while the "Oranje" thrashed Northern Ireland 6-0 in Amsterdam.
"This is a good finish to the weeks of preparation," said Dutch striker Robin van Persie, who scored his 28th goal for the national side against the Northern Irish. "We ended on a high. That is good for the mood."
Denmark's Nicklas Bendtner complained that they were too sluggish against the Socceroos but added: "We've trained extremely hard in our preparations and I'm certain the sharpness will return."
Germany captain Philipp Lahm, meanwhile, has challenged his team to start with a roar when they face Portugal in the Ukrainian city of Lviv on Saturday.
The beaten Euro 2008 finalists are among the pre-tournament favourites but suffered a set-back after a shock 5-3 friendly defeat to Switzerland and laboured to a 2-0 win over Israel.
Concerns have also been expressed about the effect of Bayern Munich's Champions League final defeat, with eight of the Bayern team that lost to Chelsea last month in the Germany squad.
Lahm said that Germany had "sensational quality" but needed to show it, adding: "It is important we hit the ground running in our first game."
Portugal travel to Poland on Monday with their morale low after losing a warm-up game to Turkey 3-1 in Lisbon on Saturday, prompting fears that they will struggle to progress.
In other groups, the holders and favourites Spain were expected in the southern Polish city of Krakow on Tuesday, after their 1-0 friendly win over China in Seville last weekend.
They take on Italy - hammered 3-0 by Russia on Friday and hit by a matchfixing scandal back at home - the Republic of Ireland and Croatia in Group C.
France, favourites for a quarter-final spot from Group D, play their final warm-up match against Estonia on Tuesday, while England have suffered a fresh injury blow.
Chelsea defender Gary Cahill was ruled out after breaking his jaw in the 1-0 friendly win over Belgium on Saturday, adding to the withdrawals of his Stamford Bridge team-mate Frank Lampard and Manchester City's Gareth Barry.
England coach Roy Hodgson later faced accusations of lack of respect from Rio Ferdinand's agent for picking Liverpool rookie Martin Kelly as Cahill's replacement instead of the veteran Manchester United defender.
England are already without Old Trafford star Wayne Rooney for their opening two matches as he is suspended.
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