They wrote letters of reference that turned out to have weight with my fellowship sponsors. Try to meet some of the journalists who come in and out of the Commonwealth for brief visits. I know it is tough for the Slavic mind to open up and tell your story straight to someone with whom you haven't eaten a ton of salt, but try. Open your heart. No one can guarantee you will succeed, but once you have a friend and colleague in the West, you can move mountains.
Friends can help arrange shorter exchanges, too.
Swap workplaces for a month or two. Several hundred newspapers in the United States might be able to do it. You may fail to get an editor of a metropolitan daily in the United States interested in the exchange by mail, but an insider friend can move and shake things. Develop human stories about your locality and bring them over. Be ready to share them with the readers.