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As Dean has indicated above, if you won't be receiving any money from Romania itself, you may be eligable for a business visa.
Read the Expat Arrival's pages on Work Permits for Romania for more detailed information. As you mentioned nationals from a whole range of non-EU countries, it might be best to approach the Romanian Ministry of Home Affairs to get more specific requirements for nationals of each country.
Hope this information is helpful.
Everyone needs a visa to work in Romania, but it will be easier for EU citizens to get a visa. Romania operates according to a quota system - there is a limited number of foreigners who are granted work permits each year. Having a job lined up is not a guarantee that a visa will be granted.