Now, feeling the pain of a 2-0 defeat in their first game as defending champions, they will have to find something else.
The team that went from upset to upset in Portugal at Euro 2004 to go home with a trophy no one outside of Greece expected to see came to Austria with a more positive approach and hopes of showing the world that it has more to offer than scrambled goals off set pieces.
After 67 minutes of play that did little to impress the neutrals in the 30,000-seat Wals-Siezenheim stadium, Sweden forwards Zlatan Ibrahimovic and Henrik Larsson showed them how the game should be played.
The lanky Ibrahimovic swapped passes with the veteran Larsson near the corner of the penalty area and drove the ball into the top far corner of the net.
It was a goal of supreme quality and something that the European Championship deserves.
Although the second goal was a scruffy, close range scramble by Petter Hansson, the Greeks have a tough road ahead if they are to hold onto their title.
After Spain's 4-1 victory over Russia in Innsbruck, Greece is already three points behind two of its big rivals in the group. Although it could well beat Russia in its next game on Saturday, it then faces the Spaniards on June 18 back in Salzburg, and that could be the day it finally says farewell to the title.
Otto Rehhagel, the German coach who masterminded Greece's Euro 2004 triumph, will have to find something different to unlock the Russian defense. Then it has to withstand the talented Spanish midfielder and strikeforce.
A repeat of this unimaginative, colorless play simply won't do.
With seven players from the victorious 2004 squad in the starting lineup, it was interesting to see whether Greece would play the same economy class football that won it the title in Portugal or whether Rehhagel would let the team maintain the impressive play through brought it to the final tournament again.
When Angelos Charisteas, whose goal in the final in Lisbon won Greece the title, went on a weaving run past three Swedish defenders as early as the seventh minute, the signs were good. Sadly, his weak shot at goalkeeper Andreas Isaksson was a prequel to what was to follow.
Although Charisteas went on a similar run later in the game and tested Isaksson with a powerful angled shot, Greece only went close when Hansson almost headed a cross into his own net.
Apart from that, Greece gave Isaksson little more to do. Its tactic of playing pass after pass along the back like presumably to try and lure the Swedes out of position simply didn't work. All it did was draw loud jeers of derision from the Swedish fans and irritate the neutrals.
Whenever they tried to deliver a long ball to the front men, it usually sailed out of play or to a Swedish defender. The team didn't use midfielder Angelos Basinas often enough and striker Fanis Gekas, who scored five goals in the qualifying games, was replaced at halftime after producing virtually no threat.
It was like watching an Italian team holding on to a 1-0 lead, not a team hoping to begin its defence of the European Championship with a victory.
Greece has a well-organized back line of tall, powerful defenders and they give the team a strong platform for Rehhagel to work on. Until the 67th minute, they stifled the dual threat of Ibrahimovic and Larsson.
But the coach needs to find a better way of getting the best out of Charisteas and Giorgios Samaras, the lively 23-year-old forward who tried in vain to get past the Swedish defenders on the few times he had the ball at his feet.
The Swedes are nowhere neat the best of the 16 teams at Euro 2008, but they knew what to expect.
"We knew they might play up to five defenders, and they would have man-markers," Sweden coach Lars Lagerback said. "It's tough to face them because they are physically strong. I'm not surprised they had one marker on Zlatan."
It was all too predictable.
"Some of my players were unable to play at the level I expected them to," Rehhagel said. "They tried hard but it wasn't enough."
The wily German clearly believes his players have the ability to give more to this competition. He needs to find it fast.
