Omagh to Dublin.

Off the top of my head, via Monaghan town, Carrickmacross, Ardee... etc seems to make sense.

A5/N2 practically all the way. Depending on where in Dublin you are going you can join the M1 shortly after Ardee or go on in the Ashbourne way.
 
The best way is all the way up the N2 through slane, ardee going from finglas should take you bout an hour and a half from finglas.
 
join the m1 at the ardee roundabout, saves goin thru ardee, slane, ashbourne, much quicker, but you need a few euro for toll
 
Taking the brief from the original question, it should be said that the M1 option is faster than the N2 route.
 
I do this drive from Donegal regularly and find this to be the best route to North Dub. (As caveat says, depends on where you're going to in Dublin.)

Omagh, Aughnacloy, Emyvale, Monaghan, (bypassed)Castleblaney (bypassed), Carrickmacross (bypassed), at Ardee roundabout take exit to M1 and then M1 all the way into Dublin.