My parents both emigrated to the US from Lithuania. I am first generation Lithuanian/American - but like you, my surname is Irish - Patrick. 
This came to be when my father was processed through Ellis Island - they Americanized his Lithuanian surname - Patrickanas.
My first name however is truly Lithuanian.
Just as an observation - the ancestral religion of Lithuania was and is the religion of Nature -  Pagan. 
It is still practiced internationally today as Romuva with witch I identify
http://www.historyextra.com/qa/religion-lithuania-pagan-conversion-christianity
The list of Lithuanian Pagan Gods is poetically romantic.I also identify with Aitvaras which is what my mother called me when as a child I misbehaved. And she was Roman Catholic. 
http://www.historyextra.com/qa/religion-lithuania-pagan-conversion-christianity
https://en.wikipedia.org/wiki/Aitvaras