The closest I usually get to a contra-style is DoSiThree.  Everybody gets the progression.  The hardest part about that dance is getting them into a 3 face 3 position -- more particularly, a set of 3 people can be a challenging concept for some (you need lines of 3 people, you and two partners, etc.).  Once they get in position, the rest of the dance works great.

I prefer longways dances instead -- Blobs is a staple and Brandy Rump Bump for a younger crowd.

Another contra dance that has worked well for me with a majority of beginners has been Moonstruck -- sliding left to progress to the next couple is easy for anybody to understand, you interact with your partner and your neighbor, and all you do is clapping, dosidos, circles, balancing and sliding to the next.

One other is French Wedding - dosidos, 2-hand turns, circles, dig for the oyster/dive for the clam to the next.

If I only have an hour, I don't call a contra.  But if I have 2 hours, well...I don't know about you, but I like just a little variety.