We came a little later than the Mayflower. I believe my dad's mother's family came over from England on the Ark and/or the Dove in the 1630s. My mom's dad's family came over from Germany in the 1700s.
Most of mine came from the 1630s as well, as that was the period of the Great Migration.
I think if folks really investigated it they might be surprised.
Millions, for example, are descended from John and Priscilla Alden (me included) of Longfellow fame (The Courtship of Miles Standish).
Priscilla Alden came over as Priscilla Mullins, the sought-after daughter of William Mullins, who died that first winter of 1621. I claim all the Mullins in my lines.
Resolved White and his wife and family are also in my direct lines.
Edward Winslow, the man who wrote the famous letter from which the story of the first Thanksgiving was rooted and who was governor as well as several other things related to the original Colony, was a great uncle.
Through my Mother's line we're related to John Howland, another influencial man of the Colony of Plymouth.
If you've got an account at FamilySearch.org there's a great site called RelativeFinder.org that helps make these connections.