E. Europe countries are now part of the European Union, and are allowed to come to the US under the Visa Waiver Program. e.g. Czech, Solavia, Lithuania, Latvia, Estonia, Hungary, and Slovenia.
It depends on, in part, if they are coming from a country where they can argue they are persecuted (and it certainly helps if they are currently stateless), what profession they are in and what sort of job they have lined up or how much money they are prepared to invest, and what relatives they already have in the USA.
Best bet is to find a good job and have a company sponsor you, I think. Best chance for that is if you have some specific skill in which languages are relevant – in my case, for example, fluent Dutch, Hebrew, and pretty good German, combined with a pretty high level of IT knowledge and experience. I actually had some opportunities in this direction but declined them because I’d rather stay in Europe for numerous reasons.
The most effective way is to overstay a tourist visa. Then lay low, and no one will bother you. And eventually they’ll be an alien amnesty, sooner or later, that’ll legalize your status.