One common misconception about the Pilgrims is that they were Puritans. They were not. About half of them were Separatists, and the rest were basically hand-selected, non-religious skilled workers.
Puritans didn't start arriving in America until about a decade after the Pilgrims arrived.
Add the Moravian settlers to the bunch, and it gets even more fun.
Regardless, this tossed salad of settlers made for an interesting musical start for America.
Let's look at one group at a time, though. The Separatists and Puritans only allowed for Psalm singing. This worked out particularly well, given that there wasn't a ton of room on the boat ride for instruments in the first place.
Separatists on the Mayflower brought over a book of psalms compiled by Henry Ainsworth, known as the Ainsworth Psalter.
Colonists also had copies of a psalm book referred to as "Sternhold and Hopkins." Yet church leaders came to feel the texts in these psalters wasn't literal enough from Scripture, so they started re-translating them and put them into a new book.
This new book, the Bay Psalm Book, was the first book ever printed in America, in 1640. There are only 11 extant copies of the first edition, one of which is going up for auction soon.
The Bay Psalm Book doesn't have any music in it; just text. We didn't have music engravers in America in 1640. Colonists actually started forgetting how to read music.
Churches started a practice called "lining-out", which meant one person would sing how the line should go, and the congregation would repeat it. As a result, the same song sounded quite different from congregation to congregation.
The influence of the Moravians, however... this is a different story. Where the Puritans and Separatists didn't allow instruments in worship, the Moravian church all but insisted on it. They had an assortment of instruments in their possession by 1642.
Hear examples of early American sacred and secular music on Learning to Listen.