Some of these countries are fragile young democracies that may become destabilized by the collapse of their economies.
Some of the governments in the young democracies are inexperienced, inept or even corrupt. Some may react in ways that threaten global stability. For example, Ukraine recently joined the WTO, but after its dramatic Q4 descent into Depression it has slapped a 13% tax on all imports. Since protectionist moves invite retaliation, this isn't a good omen.
Some of the European periphery countries are vulnerable to threats from other states - notably Russia, but the US is not averse to making power grabs where oil is produced or transported.
All of these countries have enormous foreign debt, much of which is not going to be repaid, which will cause another major hit to the global financial system and economy. Paul Krugman says that the epicenter of the world economic crisis has moved from the US housing market to the periphery of Europe. If we knew which banks and countries hold a lot of eastern European debt, we might be able to predict the next round of disasters.
Many of these countries are recent members of the EU, and many use the euro, resulting in who-knows-what down the road for Europe. The euro, which recently looked like a contender to knock the US dollar off its perch as world currency, already took a blow in this economic crisis when the failure of coordination among European central banks meant the exchange rate plummeted. More trouble is in the future for the euro and EU.
Many of these economies are highly integrated with their neighbors. For example, Latvia is going down, and is expected to take Estonia and Lithuania with it - even though Lithuania has a strong economy.
The IMF is already starting to inject money into these countries, but the collapse has just begun, and the IMF has limited funds.
Update: Latvia debt rating cut to junk