However, this five-year period of success was followed by failure to qualify for the next two international tournaments, before eventually qualifying for Euro 2008 in scrappy style, pipping Norway by one point.
Turkey went into that tournament managed by the great Fatih Terim - arguably the most important and influential figure in the history of Turkish football - yet expectations were low. There were huge question marks over their defence, and their style of football in general. Even Terim was unsure about his best team leading into the tournament.
They did possess some attacking quality though. Nihat Kahveci had scored 24 goals for Villarreal that season, while the supremely talented Arda Turan was beginning to burst onto the scene.
Comebacks galore
Turkey were drawn in a group with Portugal, joint-hosts Switzerland and the Czech Republic. Their campaign got off to a slow start, falling to a 2-0 loss against the Portuguese, without putting up a fight.
But the drama was yet to begin.
Switzerland were up next, and following some torrential rain, the pitch was totally waterlogged which led to a really scrappy, ugly contest.
The Swiss took the lead in the game after a puddle in the penalty area slowed down a cross to allow Hakan Yakin a simple tap-in.
Turkey were not playing well. The issues that everyone was concerned about prior to the tournament - the poor defence and lack of direction - were on show.
Switzerland continued to create chances, but were letting Turkey off the hook.
And they were duly punished. Around 10 minutes into the second half, Semih Senturk headed home.
The game looked to be petering out to a draw, which would have been a poor result for both teams following losses in their opening games.
Yet in the 92nd minute, Arda saved the day for Turkey, smashing in a deflected effort leading to insane celebrations on the Turkish bench.
They had saved their Euro 2008 campaign by the barest of margins.
"Euro 2008 has started for us today," Terim proclaimed after the match.
Going into the final group game, Turkey and the Czech Republic were level on points, goals scored, and goals conceded, so a draw between the pair would have led to a penalty shootout.
Turkey weren't thinking about that. A win in 90 minutes was the only option for them.
But their dream quickly turned into a nightmare.
In the final match, the Czechs took the lead after half an hour courtesy of Jan Koller, before doubling their lead on the hour mark thanks to Libor Sionko.
It was all going wrong for Turkey. They weren't playing well, their defence was being carved open, and the Czechs were smelling blood and in full control.
However, with 15 minutes to go, Arda pulled a goal back for the Turks, finishing calmly off the post. The comeback was on.
And disaster struck for the Czech Republic. With three minutes left of normal time, Petr Cech fumbled a cross, spilling the ball into the feet of the grateful Nihat, who tapped into a gaping net.
Penalties looked to be on the cards, something Turkey would have snatched your hands off for with 20 minutes left of the game. But the momentum was now with them. They could sense a miracle.
Just two minutes later, Nihat was played through on goal and slotted the ball home with stunning composure. One of the greatest comebacks in European Championships history was complete. The bench were euphoric.
Turkey, despite stuttering and struggling for all three of their group games, had somehow found a way to qualify.
The drama wasn't over in the game though, with goalkeeper Volkan Demirel getting himself sent off. With no substitutions left, attacking midfielder Tuncay Sanli had to see out the dying minutes in goal, which he did successfully.
Turkey were back in the knockout stages of a major international competition. And their opponents in the quarter-finals were Croatia.
The Late Show
Croatia were the heavy favourites. They had topped a group with Germany, Austria and Poland in it, picking up nine points from nine. Observing how both teams were playing, many feared it had the potential to be quite one-sided.
Turkey and Terim were also enduring a major crisis within their squad. 35-year-old Rustu Recber had to come into goal following Volkan's red card, while a number of other players were injured, including the crucial cog Mehmet Aurelio. Overall, they had just 15 players fit and available for the quarter-final.
The game didn't go as expected though. Croatia controlled it and created the clearest chances, but it wasn't as dominant as everyone had thought.
After a pretty dull 90 minutes, the match went to extra time, where the drama really began.
In the 119th minute, Rustu got caught out, giving Croatia a simple chance and Ivan Klasnic headed into an open net.
The game was surely done. Turkey couldn't do it again, could they?
One minute of added time was awarded. Turkey were throwing everyone forward. The clock had gone past the allocated minutes.
Rustu launched the ball forward, hoping for one final miracle. And they got it.
The ball fell kindly for Semih Senturk, who volleyed home in emphatic style to rock the stadium and break the hearts of Croatian fans. Another late show. Another goal in the dying embers of a game to save their tournament. They'd done it again.
The fact of the matter is, once the game had gone to penalties, there was only one winner. Croatia were deflated and looked defeated already, while Turkey had a spring in their step. They had a lifeline.
Turkey scored every single one of their penalties, while Luka Modric and Ivan Rakitic missed the target entirely. Mladen Petric's spot kick was also saved by Rustu, sending Turkey through to the semi-finals.
They were the miracle men. Fate and fortune were on their side. Everyone was captivated by the stunning run of the Turkish national team.
“We seem to come back from the dead, which shows you how good my team is,” Terim said.
“I saw some of the players lying on the pitch. I told them to pick themselves up. I wasn’t about to give up. I never have in my career. You should never give up until the referee blows the final whistle.”
The Dream Ends
Their semi-final opponents were Germany. If Croatia were heavy favourites in the quarter-finals, the Germans were overwhelming favourites to secure a place in the final.
Turkey's squad crisis had gotten even worse too. They now had just 14 players available, with Rustu still in goal.
The game began, and for once, Turkey were playing well. They looked slick and up for it. They were on top in the game, and found themselves in a position they weren't used to: leading after a goal from Ugur Boral.
But the lead didn't last long. Just three minutes later, Bastian Schweinsteiger equalised for Germany. Normal service resumed.
Yet Turkey were looking good. It was the best they had played all tournament. Against all the odds, they were giving Germany a proper game.
Turkey continued to create chances and pepper the German goal. However, against the run of play, Miroslav Klose put his country ahead. With around 10 minutes left, Rustu made yet another error, leaving the goal open (again) for the striker to score.
Still, as they had done all tournament, Turkey weren't done. Semih was the hero once more, poking the ball past Jens Lehmann to make the score 2-2 with four minutes left.
The Turkish miracle was on again. The belief instilled within the team by Fatih Terim meant they never knew when they had been defeated.
Unfortunately, sometimes you need a little bit more than belief.
In the 90th minute, Phillip Lahm finished emphatically to put Germany 3-2 up and win the game. Turkey had got a taste of their own medicine. They were going home.
Remarkably, Turkey were the better team. They had 11 shots on goals compared to Germany's three. And they lost the game.
On the other hand, in the games in which they were outplayed by Switzerland, the Czech Republic and Croatia, they won. It's funny how football works.
And how differently it could have panned out had Volkan been in goal in place of Rustu. How differently it could have panned out had they not endured an injury crisis. I guess we will never know.
But they had captured the hearts and minds of football fans around the world. It was one of the most miraculous runs in the history of the European Championships. Throughout the tournament, they had led for just a staggering 13 minutes.
Terim would step down as manager of Turkey following the competition. He had done a fantastic job of overachieving with a side which had looked in trouble just months before. His squad and man management had been second to none.
“I simply told the players that I am proud of them and that they should not feel bad about the result. I wished them a lot of success in the future," he said.
"Seventy million people and their hearts were beating with the players tonight; that is very important. We have been able to show to the world what a good team we are.”
Football is often seen as purely about winning silverware and lifting the most prestigious trophies. But occasions like this suggest there is far more to the sport than just that.
The years leading up to the tournament had seen a bit of a disconnect between the national team and the fans, so this run finally brought back the same feeling that they had felt in 2002. In fact, this probably meant more.
The results in 2002 were almost expected as it was their golden generation. But in 2008, they defied all the odds and logic to reach the semi-finals. So the desire and never-say-die attitude on show was a more breathtaking experience than six years earlier.
The players and manager had given them a reason to believe. The passion had returned to a group of people who love football as much as anyone on earth.
Memories were created forever as they made an unforgettable impact on the European Championships. Following the failure to qualify for previous tournaments, pride was the overriding emotion. And sometimes, that is just as important as anything else.