Croatia, which faces Turkey, has -- together with Serbia, which has failed to progress beyond the group phase -- been almost a regular atop international sports such as football and basketball. Bosnia-Herzegovina and Montenegro, on the other hand, are hoping to secure their first showings on the international stage.
Poland and Ukraine are jointly hosting the European Championships.
Bosnia faces favorites Portugal, led by Real Madrid star Cristiano Ronaldo, who two years ago denied the team a chance to travel to the World Cup in South Africa.
Next year, the country will mark 20 years since gaining its independence, which was followed by a bloody war and a volatile decade and a half of peace marred by ethnic divisions and tensions. It could hardly receive a better present for the anniversary than its first participation in the European Championships after eight attempts.
Montenegro, a sports-crazy nation of 625,000 that was drawn against the Czech Republic, is even younger, having declared independence from a union with Serbia in 2006. It sees its qualification as yet more evidence that it was right in deciding to go it alone.
In both countries, however, a large percentage of the populations -- in Bosnia, probably more than half -- do not consider the national team as their own.
Bosnian Serbs and Croats, which together make up just over half of about 3.8 million citizens, support their "homelands" -- Serbia and Croatia, respectively -- for whose national teams many ethnic Serb and Croat athletes from Bosnia choose to play almost by default.
Still, despite all the problems, such as ethnic-based fan violence and political interference in how football is run, a joint soccer league has survived, with teams from all ethnic communities and, very often, with ethnically mixed rosters.
Probably more than in any other sphere of life, sport -- and the national soccer teams, in particular -- go against the tide and genuinely unite three ethnic groups, at least on the pitch if not in the stands and in front of television sets.
Jiri Plisek, a Czech who coaches FK Sarajevo, summed it up nicely in last weekend's article by Ed Vulliamy, "The Guardian's" veteran Balkan watcher and a friend of Bosnia:
"In this country, if the politicians had their way, Bosniaks would have to pass to a Bosniak, Serbs to Serbs, Croats to Croats. But that's not how football works; football connects everyone, and has the face of every nationality. If they beat Portugal, these Bosnian boys will show the world what their country can do, that their talent can be used in a way the country should follow."
Bosnia's skipper and most capped player is Zvjezdan Misimovic, a Bosnian Serb born in Germany into a family of immigrant guest workers. He now plays for Russia's Dynamo Mosow.
The biggest star is Edin Dzeko, a Bosniak Muslim born in Sarajevo who recently joined the English Premier League's newly rich powerhouse Manchester City after a highly successful stint in Germany, where he and Misimovic conquered the Bundesliga two years ago with Wolfsburg.
Sasa Papac of Scottish giants Glasgow Rangers is a defensive stalwart who comes from the staunchly Croat Herzegovina region in the south. His celebration of the Scottish title earlier this year while waving a Bosnian flag earned him a legion of new fans back home.
True enough, Muslims dominate the team and the head coach is Safet Susic, Bosnia's best player of the past 50 years, but his coaching staff includes Serbs and Croats, too.
Misimovic says success at the tournament will mean a lot to ordinary Bosnians, a way to overcome the traces of war that still linger.
"That would be a chance to forget -- at least for a moment -- everyday problems and bring smiles back on people's faces," he says. "That is our dream."
In Montenegro, the divisions are not that deep, but in some parts of the country football allegiance is pledged to Serbia, not Montenegro.
Just above one-quarter of the population declares themselves as Serbs, against 45 percent for Montenegrins, and most of them probably voted against independence 5 1/2 years ago.
The Montenegrin team is even more ethnically mixed than the Bosnian team, as it features Montenegrins, Serbs, Bosniaks, and Albanians. Its two stars play in Italy -- Mirko Vucinic in Juventus and Stevan Jovetic in Fiorentina.
Andrija Delibasic of Spain's Rayo Vallecano says his team wants to make the most of an opportunity that doesn't come around very often.
"We will give our all and make the most out of it," he says.
-- Nedim Dervisbegovic