This is from Ryanair's web site:
All passengers must present valid photo identification at check-in for all flights. The only acceptable forms of photo-ID on Ryanair flights are:
A valid passport.
A valid Government issued National Identity Card issued by a European Economic Area (EEA) country. Only the following countries from the EEA issue National Identity Cards which are acceptable for carriage: Austria, Belgium, Finland, France, Germany, Greece, Cyprus, Czech Republic, Estonia, Hungary, Italy, Liechtenstein, Lithuania, Luxembourg, Netherlands, Malta, Poland, Portugal, Slovakia, Slovenia, Spain, Switzerland.
A valid driving licence with photo (in the case of UK domestic flights and UK-Republic of Ireland-UK routes only)
It looks as though, for travel from the UK to the Republic of Ireland, all you need (whatever your nationality) is a driving licence.