Going to the Aran islands is an alternative:

http://www.flickr.com/photos/mksfca/4172435070/in/set-72157622843664921

This just came out from Budget Travel on the best sites to see in Ireland.

http://www.budgettravel.com/feature/budget-travel-vacation-ideas-irelands-10-best-attractions,12827/

Take it with a large grain of salt. T2P thinks it's a tourist trap.

I'm going to the trap in August. Shame on me.

Our B & B host recommended the Sky Road in Clifden. This was one of the most surprising and memorable activities we had in Ireland. Clifden is Northwest of Galway. Be sure to take the upper Sky Road as the views are stunning.

We did this in one day.
http://www.mydiscoverireland.com/blog/2008/08/08/clifden-and-the-sky-road/

If you continue a little farther, you can see this much photographed Abbey. Didn't spend much time there, but it is very pretty against the lake. https://plus.google.com/114733292598385347789/about?gl=us&hl=en

We really enjoyed this National Park with the newborn spring lambs. http://www.connemaranationalpark.ie/

For the record, we did go to the Cliffs of Moher. Glad we did it but probably because I think we'd have been disappointed if we didn't see them. They were beautiful, but quite touristy. I wouldn't feel the need to go back, but I would do the Sky Drive again in a second!

I meant to say, we did ALL of that easily in one day! We were there in April too. Chilly and wet, but lambs everywhere!

The Cliffs are Tourist central and over hyped these days, unfortunately we knew them as a natural feature befor they imported a few thousand tons of hilside to make telly tubby land, sorry the visitors centre. That said Yes they are worth seeing especially after closing in the evening, find one of the few non yellow line restricted parking spots and the Public Right of way to the Cliffs. Alternatively see them on a mini cruise from sea level.

Alternatives would be Loop Head and the Cliffs of Kilkee including the Bridge of Ross But that is south.
Inis Mor would be a good day trip with a few Cliffs, the Connemara trip could include the N59 route out and some of the loop http://ireland.activeboard.com/t50055177/n59-galway-route/ includes the Sky Rd and http://goo.gl/maps/L5ihu

We took the Cliffs cruise that Tony mentions and had a great time. Much different experience than viewing them from the top.

The cliffs are spectacular on a sunny day and won't be as crowded in April as, say, August. We went the first time in 95 with our teenaged sons who thought nothing of sitting on the edge with their feet hanging over. Back then you just stepped over a little split rail fence and you were at the edge.

We went again a few years later on a foggy windy day, kept back by new walls. It wasn't the same experience.

We like to take the drive out to Loop Head in Clare. The cliffs aren't as tall but they are still pretty amazing and they are free. It takes half a day pretty much. DH loves to drive up to Slieve League in Donegal for a literal good high.

Thanks everyone for all the posts! I'm definitely going to try out Sky Road and maybe see what my husband thinks about going out to the Aran Islands

The Aran Islands are a great day trip and an even better overnight trip. Good views of cliffs, but what is even more amazing is Dun Aengus and the other ancient sites on the island.

We also opted out of seeing the Cliffs of Moher and enjoyed our morning doing the coastal walk along the cliffs in Kilkee. It was just us and 3 moms with strollers--definitely NOT touristy.

We did Loop Head / Bridges of Ross that Tony mentioned albeit on our way up from Dingle.

We also took a day trip to the Aran Islands. Liked them both.

Have a look (towards the ends):

http://ukfrey.blogspot.co.uk/2011/08/ireland.html

Achill Island has some pretty nice ones, too.

Achill depending on your definition of Sea Cliffs has the highest in Europe.
http://www.irelandtravelkit.com/highest-sea-cliffs-in-europe-achill-island-co-mayo/
All along the North Mayo coast you will find natural features not listed in many if any guide books. Downpatrick Head, Benwee Head and Belderg. However these are all a long way North from the Clare coast.