Cromwell represents an interesting representation of what the colonists both loved and feared. On one hand, Cromwell overthrew the king and succeeded in bringing about his revolutionary vision. On the other hand, Cromwell was undoubtably a despot and dictator, the very thing the colonies claimed to be resisting.

Opinions seem to vary Founder to Founder, but most seem to have a relatively positive opinion of Cromwell. Obviously this was not universal. Alexander Hamilton specifically refers to Cromwell as a despot in the Federalist Papers No. 21. He was speaking in the context of "despots" having the potential to hijack revolts and use them to gain dangerous power. It's safe to say that Hamilton was not fond of Cromwell, or at least not fond of how he ruled or came to power.

Jefferson and Adams seem to hold a slightly higher opinion of Cromwell, or at least of his revolution. The pair visited the site of Cromwell's great victory at Worcester and were greatly disturbed by the locals' lack of appreciation of history. John Adams later wrote:

"The people in the neighborhood appeared so ignorant and careless at Worcester that I was provoked and asked 'And do Englishmen so soon forget the ground where liberty was fought for? Tell your neighbors and your children that this is holy ground, much holier than that on which your churches stand. All England should come in pilgrimage to this hill, once a year."

He does not mention Cromwell by name, but clearly sympathizes with the English fight to overthrow Charles I.

The Connecticut legislature authorized the 1776 purchase of two ships to help create a nation navy, naming one of them the Oliver Cromwell.^1 This was among the first ships ever to serve in the US Navy. Clearly a significant portion of the colonies at least respected Cromwell's revolutionary achievements as they attempted to similarly rid themselves of monarchial rule.

While the Founding Fathers may have held varying opinions on Cromwell, as a group they seemed to hold his revolutionary spirit and accomplishments in esteem while holding his despotism in contempt.

^1 The Navy of the American Revolution: Its Administration, Its Policy and Its Achievements, by Charles Paullin

A potential addition to this question, should anyone feel like answering: did citizens of the 13 colonies in the 18th century consider themselves to have a direct line, through The Mayflower and The Pilgrims, to the Puritans and England under Cromwell/ The Protectorate?

ETA: I was wrong; see below

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