"I'd love to trace my Irish mother's family back (to prove she was descended from Irish kings!) "
You shouldn't have any problem with that Sen - didn't you know everyone in Ireland can be traced back to Irish Royalty!!
Tracing Irish roots is, I believe difficult, (I've never tried, but some of my family have) largely due to the mass exodus in the mid nineteenth century and the destruction of The Four Courts, where the records were held, in the Civil War.
Have you any idea where they were from - parish records seem to be a good start rather than some of the Heritage firms.
I must say, I was always impressed with the old 'Roots' book and serial, but watching the first episode of the new one last week knocked my socks off
Jim Carroll