Maybe what we need is a Stillness Movement. The goal of this, as opposed to mindfulness, is to simply do nothing. Don't meditate, don't use your computer, don't try to clear your mind, don't talk to anybody, don't do any work, just be. You'll probably be sitting, maybe lying down. But really, just try to hang out in your own self.

There is some precedent to this, I think. At least according to an episode of Six Feet Under I saw, the Quakers have something similar:

Unprogrammed worship is the more traditional style of worship among Friends [Quakers] and remains the norm in Britain, Ireland, continental Europe, Australia, New Zealand, Canada and parts of the United States. During an unprogrammed meeting for worship, Friends gather together in "expectant waiting" for divine leadings. Sometimes a meeting is entirely silent, sometimes quite a few people speak. Meeting for Worship generally lasts about an hour.

When they feel they are led by the spirit a participant will rise and share a message (give "vocal ministry") with those gathered. Typically, messages, testimonies, ministry, or other speech are not prepared as a "speech". Speakers are expected to discern the source of their inspiration -- whether divine or self. After someone has spoken, it is expected that more than a few moments will pass in silence before further Ministry; there should be no spirit of debate.

Unprogrammed worship is generally deemed to start as soon as the first participant is seated, the others entering the room in silence. The Meeting for Worship ends when one person (usually predetermined) shakes the hand of another person present. All the members of the assembly then shake hands with their neighbours, after which one member usually rises and extends greetings and makes announcements.

I certainly want to try a Quaker service. They sound very radical in a way that doesn't fit the stereotypical look of radicalism.