Most of these roads are windy, twisty (bumpy) roads, not smooth National roads and dual carriageways! I drove all around Ireland for 15 years as a coach driver and these are some of the roads that I like to drive in a car when I get the chance, including going 'the long way' to wherever I'm headed
Dublin / Wicklow:
Try the Old Military road all the way from Stocking Lane to Rathdrum via Laragh. Maybe an hour at a decent pace but without wrecking the car. Its similar to the aforementioned Hollywood - Glendalough trip but is 'the road less travelled'. Carry on after Rathdrum to Avoca if you're out for the day and you can continue to Arklow on windy twisty roads, before jumping on the N11 for a more relaxed trip back to the smoke.
Another one in the same general area starts with the Kilternan - Enniskerry road and onto the R760 past Powerscourt House and Gardens. Stay on that (local) country road until Glebe then turn right a little bit past Glebe towards Ballybawn / Ballyinteskin / Stoney Pass (or you could continue to the R755. However, while its a nice enough drive, its also the direct route form the N11 to Glendalough so gets quite a bit of traffic, large and small). You'll eventually arrive at Roundwood. Continue to Annamoe (or Ath Na mBo). If you don't want to head to Laragh / Rathdrum, or Laragh / Glendalough, turn left just before Annamoe onto the R763 to Ashford. From there, take the N11 northbound for a speedy return to Dublin.
In either case, if you don't like the N11/M11 return, head East of the motorway and look for the R761 up through Newcastle, Kilcoole, Delgany, Greystones, Bray etc.
Mayo / Galway:
Try the Famine Trail, the R335, southbound from Loiusburgh down to Ashleagh Falls. About 30Km. Add in another 20Km if coming from Westport (a stretch of road I don't like - no idea why - but has to be done to get to Croagh Patrick / onto Louisburgh). Continue West on the N59 to Kylemore Abbey / Letterfrack. I dont like the loop road continuing westbound from Letterfrack via Clifden, but thats cos I drove it a few times in full size coaches and its very long, its not comfortable and it doesn't really offer anything 'different'. Alternatively, continue southbound on the R336 down to the N59 Galway-bound. Turn left onto the N59 heading East through Oughterard and onto Galway City if you want. If you're out for a looonnnngggg day, cross the N59 instead, at Maam Cross, and continue south on the R336 into the Gaelteacht areas. Don't be tempted to do the loop to Rosaveel. It doesn't offer much else (except the Aran Island ferry!) but continue onto Inverin, Spiddal etc and into Galway City. While the 'long' journey isn't much in terms of miles covered, it will take 2-3 times longer than you might expect
In North Mayo, for a strangely straight and flat country road, try the N59 northbound from Mulranny. See if you can locate the 'musical bridge' on that road (try Google!). Continue on the N59 to end up in Ballina, Mayo (I don't remember much about the second half of that road - the top, eastbound, part of the loop. I've only been on it a couple of times). For an alternative, and longer, trip, at the top of the northbound part turn westbound onto the R313 at Bangor Erris and head for Belmullet. There's not a huge amount to see / do there but... well, you're not likely to be in the area very often so you might as well if you have the time!
Clare:
I like the N67 from Kilcolgan to Ennistymon via the Cliffs of Moher. Turn onto the R478 at / after Lisdoonvarna for the Cliffs road. A few straights, plenty of tight bends, a nice set of S-bends up the hill to the Cliffs, varied scenery and a few things to stop at along the way.
For a bit more of an 'away from it all' feel, try the R480 south of Ballyvaughan, stopping at Poulnabrone Dolmen along the way. Worth a one-time visit if you're in the area. Continue on to Kilfenora (of the 'Kilfenora Ceili Band' fame), then West towards Lisdoonvarna to continue towards the Cliffs. This route will be longer, and you'll miss those uphill S-bends so its a matter of personal preference. Unless you're going to return the same day, then you can do both. If so, do Poulnabrone on the outbound trip and the S-bends will be downhill on the return.
Kerry:
The obvious one is the N70, Ring of Kerry. Drive it anti-clockwise if you don't want to keep coming face-to-face with tourist coaches or drive it clockwise if you don't want to get stuck behind them *. Alternatively, drive it late in the afternoon if you want to (mostly) avoid them. It might be a cliche of a trip but it is still worth seeing. The variety in both landscape and road types will make it interesting. Allow 4-6 hours (although its only about 100Km) so you can stop plenty of times. Start after 11am if you want to let the tourists get at least a decent bit ahead of you (assuming you start in Killarney. Tours usually start between 8am and 10am). Take a short detour up to Aghadoe Heights for a really impressive view.
(*)Tourist coaches are generally advised to all drive it in the same direction (clockwise) so they don't come face-to-face on a narrow bendy bit of the road, although its mostly pretty good these days. Plenty of things to see & do, plenty of places to stop for a break.
Limerick:
The R496 then R493, starting at Birdhill, via Killaloe, Northbound along the lakes to Tuamgrany, then Scarriff/Tulla, then the R352 to Portumna. Its about 80Km but its a slow 80Km! Allow a couple of hours. Not much to stop for along the way, except the odd pretty town (eg Killaloe. Lovely setting. Worth parking up and getting out of the car to explore as otherwise you'll just bypass it)
Happy to provide some more suggestions if anyone wants to give me a shout with a specific area in mind...