I love to go running every Saturday morning, and my favourite part of the city for this is the seafront in Clontarf. It's got just what I like for jogging - ground not too hard or soft, fresh air, and nice wide open views.
Herbert Part is quite nice as well, but a bit cramped compared to the bright, breezy seafront.
Dub Laoghaire east pier is good for walking but personally I prefer not to run on such a hard surface.
Phoenix Park... well, wow, what's a whole other subject. We're lucky, lucky, lucky that it didn't get sold to a property developer!