You will enjoy touring Ireland, enormously, people are great (generally - see observation below on bike lifetimes in Dublin), beer is great, countryside very variable, weather can be a little on the iffy side, come prepared for all weathers.
Now, one caveat, I am not sure whether the lifetime (from the owners point of view) of a bike is shorter in Dublin or Amsterdam. One way or another, it 'aint long. You must, especially if you ride anything other than a clunker, find somewhere secure to store the bike while you stay in Dublin. Public transport is pretty good there (at least compared with just about anywhere in the US) and you can probably do without your bike while you are in Dublin itself. If you need some work done on your bike while you are in Dublin (or just advice and a chat) go and visit SquareWheels, great cafe just above and very close to the center of the city (which is mainly walkable, public transport making up the rest).
Once you get out of Dublin, not quite the right direction, but you will find Newgrange (a bronze age burial site within easy riding distance) well worth the visit. From then on, Ireland is your oyster.
If you are trundling over to the west coast, make sure you bring your granny (gears)! Don't neglect the North if you have time, some of the most stunning scenery is in Antrim (and the toughest cycling).
Make sure you come prepared for most weathers (even in the summer). Although the extremes do not match the US, you can expect it to alternate between comfortably warm and wet'n'miserable (ok, just wet then) on occasions. At the same time, don't overburden yourself with gear. We tend to to tour "credit card" style (although without the credit card balances to match), if you just have a couple of small panniers on your bike, it makes life a great deal easier. You can always do without the kitchen sink, and the scubbers and....
As for places to stay, we have always used a combination of bed and breakfast (cheap, and normally very good) and youth hostels, including the "Independent Youth Hostel Association" (possibly not the correct name, but the hostel group independent of the mainstream). The Independents are often based in some of the large (and I really mean large) country houses which are now too big and expensive for one family to maintain. They also often have other small businesses (we came across organic farms, gourmet restaurants, healing centers... you name it its there). Great fun to stay in, even if sometimes a little spartan. We have always been made to feel welcome. In a number of cases, it is the same family who have owned the house for 100+ years and they have a great, if decayed air about them.
Guides : despite minor problems, I can not but recommend "A rough guide to Ireland" - far far better than the Lonely Planet Guides (their nearest competitor imho) and an infinity away from the glossy, picture rich guides that give you little useful information.
Enjoy yourselves, Ireland is a great place to travel.
-- ,_~o Dr. Richard Taylor email : email@example.com _-\_<, Computer Systems Engineering tel : (616) 387 4826 (*)/'(*) WMU, MI 49008