Funny you should mention it . . . .
I was on Kauai in April 2009, and rented a road bike for a day from these nice folks. http://kauaicycle.com/index.html
They seemed to be about the only shop that had real road bikes to rent. They are in Kapa'a (just south of the main downtown section) and took good care of me. Warning - If you are a big guy (like me), you will be cramped - I think the biggest road frame was a 58 or 59cm. I take a 63, give or take, and I baaaaaarely made it work. Definitely bring your own pedals and shoes. I would also suggest contacting the shop to find out what to expect in terms of gearing. And do not expect anything very new - I was on a Specialized CF bike of some sort with Shimano 8-speed - it worked fine, but it had definitely seen its better days.
I rode north on the main road from Kapa'a, with one short side trip up Kealia Road (the southern end is rolling and pretty and the road is good, the northern part is rolling and pretty and the road is terrible), and I am not in a hurry to do that again. Plenty of shoulder, but also plenty of high-speed traffic.
As for riding itself, it is mostly so-so but with some exceptions that could be glorious if you can get there. If you look at a map of Kauai, you will see that there is basically one road that goes most of the way around the island near the coast (there is no road on the west side - heck, there isn't even a trail that goes through there, it's that rugged). Anything and everything else is an offshoot of that main road. And that main road, while wide enough to be safe for the most part, has enough traffic to make it unfun, at least between Hanapepe and Princeville. From Princeville out to the end of the road on the northwest end of the island is very pretty, but the road narrows and the drivers can be kind of nuts, so go at a low-traffic time if you choose that one.
If you can, and you don't mind climbing, ride the Waimea Canyon road. Kokee Road (hwy 552) is the easier way up the first part. A ride from the bike rental place in Kapa'a to the top of the Waimea Canyon Road would be an epic ride, but getting fromn Kapa'a to Lihue would suck, traffic-wise, and from Lihue to Hanapepe would be only somewhat better. There is some time off the main road available around Poipu, but not for long. I would put the bike in a car, drive to somewhere between Poipu and the base of the canyon, depending how long you want to be on the bike, a go from there. One way or another, though, you have to see Waimea Canyon - simply glorious.
Another fun road, but probably too short for you, is Kuamoo Rod (hwy 580). It takes off from the main road between Lihue and Kapa'a right next to the Wailua River. It climbs past a pretty waterfall, goes through a pleasant residential area, and eventually (as in 5 miles or so) ends up at a trailhead where the road fords a stream and soon becomes dirt. No shoulder on this road, but the traffic is not too bad.
BTW, keep in mind that the northern side of the island is significantly wetter than the southern side, at least along the coast. Inland gets wet in a hurry pretty much everywhere. But it can be raining in Princeville and bone-dry in Hanapepe.
Have a great time. And be sure to get to Waimea Canyon, even if it isn't by bike. You will not be sorry.