Easy way to prevent Heroku idling?

You can install the free New Relic add-on. It has an availability monitor feature that will ping your site twice per minute, thus preventing the dyno from idling.

More or less the same solution as Jesse but maybe more integrated to Heroku... And with a few perks (performance monitoring is just great).

Note: to all those saying it doesn't work: the important part in my answer is "availability monitor". Just installing the addon won't help. You also need to setup the availability monitoring with the URL of your heroku app.

As an alternative to Pingdom I suggest trying Uptimerobot. It is free and offers 5 min interval site checking. It works very fine for me.

UPDATE 7th of May 2015: This will not be possible any more, as Heroku will change their free dyno to prevent keeping it alive for full 24 hours:

Another important change has to do with dyno sleeping, or ‘idling’. While non-paid apps have always slept after an activity timeout, some apps used automatic pinging services to prevent that behavior. free dynos are allowed 18 hours awake per 24 hour period, and over the next few weeks we will begin to notify users of apps that exceed that limit. With the introduction of the hobby dyno ($7 per month), we are asking to either let your app sleep after time out, or upgrade to this new option.

When is this going to be live? According to their blog post:

Applications running a single 1X dyno that don’t accumulate any other dyno charges will be migrated gradually to the new free dynos beginning on July 1.

You can also try http://kaffeine.herokuapp.com (made by me), it's made for preventing Heroku apps to go to sleep. It will ping your app every 10 minutes so your app won't go to sleep. It's completely free.