The greatest demand for teachers is in the fast growing metropolitan areas, particularly the countryside around Dublin. And the government is pumping money into underprivileged areas/schools, which happens to be where teachers are doing their best to get transferred out when burnout sets in.
Outside of Dublin, there are jobs available, but the system and style of teaching is unique (teach to the test!) so be prepared for a huge mental shift if you've been a creative genius in some American high school.
Third level, college jobs face the same hurdles but the universities are used to dealing with the paperwork and applying for such jobs. Unfortunately, it's in the sciences that there's the demand. However, there's at least some greater hope there.
One area that's outside the normal quotas and mandates is special needs. If you hold a credential in some language related area of special needs (speech teachers, etc) then there would also be more hope.