After beating Georgia 3-1 to get their tournament off to a flier, Turkey were swatted aside 3-0 by the class of Portugal.
Montella's men are one of the best supported sides in Germany, where nearly three million people with Turkish roots call home.
However, there has been a backlash to the Italian boss and his squad after failing to compete with Portugal for large spells in Dortmund.
"I'd like to remind the critics, if you love this country, support your country as much as you can," Montella told reporters at his pre-match press conference on Tuesday.
"This is the only team that represents Turkey. These lads have put a smile on our face in recent teams.
"We need your support and the players deserve it based on what they have achieved on the pitch."
Turkish football features some of the fiercest rivalries between clubs in the world.
But Fenerbahce winger Irfan Can Kahveci urged fans to put those tribal differences aside to unite in support of the national team.
"Within the team we have a great atmosphere and we try to demonstrate that whether the players are from Galatasaray, Besiktas, Trabzonspor or Fenerbahce.
"We are united under the national flag. We should let those divisions go."
Turkey remain second in Group F and just need a draw against the Czechs in Hamburg to secure a place in the last 16.
Even a narrow defeat could see Turkey through as one of the four best third-placed sides.
But Montella said he would set out for victory.
"We have to go out and win the game," he added. "We can't get hung up on permutations."
Real Madrid's Arda Guler is set to return to the Turkish starting line-up after Montella revealed he was only left on the bench against Portugal due to an injury that meant he could not play the full 90 minutes.
"If he doesn't play it will be footballing reasons," said Montella. "He didn't start the last game not because of footballing reasons."
