In a letter to his friend, mentor, and former professor (from his days at William & Mary College), the Scotsman and an, by virtue of his teaching of Jefferson, important evangelist of the ideas of the Scottish Enlightenment in American, William Short, Thomas Jefferson sums up his interpretation of the Epicurean philosophy:

Syllabus of the doctrines of Epicurus

Physical. – The Universe eternal.
It’s parts, great and small, interchangeable.
Matter and Void alone.
Motion inherent in matter which is weighty and declining.
Eternal circulation of the elements of bodies.
Gods, an order of being next superior to man, enjoying in their sphere, their own felicities; but not meddling with the concerns of the scale of being below them.
Moral. – Happiness the aim of life.
Virtue the foundation of happiness.
Utility the test of virtue.
Pleasure active and In-do-lent.
In-do-lence is the absence of the pain, the true felicity.
Activity, consists in agreeable motion; it is not happiness, but the means to produce it.
Thus the absence of hunger is an article of felicity; eating the means to obtain it.
The summer bonum is to be not pained in body, nor troubled in mind.
i.e. In-do-lence of body, tranquility of mind.
To procure tranquility of mind we must avoid desire and fear, the two principal diseases of the mind.
Man is a free agent.
Virtue consists in 1. Prudence. 2. Temperance. 3. Fortitude. 4. Justice.
To which are opposed, 1. Folly. 2. Desire. 3. Fear. 4. Deceit.

His description of how the gods interact with humanity does not just reflect the ideas of Epicurus, as we know them, but also deism (which, I would argue, reflects the beliefs of Jefferson and Washington, at least, among the Founders; though it is not typical of the mostly staunchly protestant thinkers of the Scottish Enlightenment, but rather of the French Enlightenment; of course, that greatest of all figures of the Scottish Enlightenment, David Hume, was almost certainly atheist).