Irish?! "top o' the Mornin' t'ye Thoth Amon!" or "Why yis, ifn' it be bleedin' it con doy!" Orrrr "Jaysus and begora, lass, tis a dragon!"
Welsh: No way! Why? Because his name is Conan, not Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch.
The historical Cimmerians were an Iranian tribe, so he'll have an Iranian accent, or possibly Armenian given where they ended up.
Now I can't get the idea that he sounded like Freddie Mercury or Serj Tankian out of my head
Maybe you're thinking Summerians?
The Cimmerians of Howard were definitely more like the Celts geographically, or even the Picts, at the very least closer to Western Europe than the Iranians.
OP: To answer your question, Conan sounds almost exactly like Ron Perlman or, failing that, Michael Donovan from the cartoon "Conan the Adventurer". However, he looks exactly like Jason Momoa with maybe a few more pounds on him.
If I had to pick a modern day accent I'd say American Fake Barbarian voice. Lots of diaphragm and over-pronunciation, some grunting and guttural accentuations, with otherwise clear vocal tone. Think of it this way: their god's name is "Crom", his name is Conan. Deep O's and Ah's, not "Wheedly deedly doo".