You could use LINQ:
Assert.That(people.Any(p => p.Name == "joe"));
or, if you want to be explicit about there being exactly one person with each name:
Assert.That(people.Count(p => p.Name == "joe"), Is.EqualTo(1));
If you want a better error message than “Assertion failed, expected true, was false”, you could create your own assert method.
For several collection-related asserts, CollectionAssert is very useful – for instance, it allows you to check if two collections contain the same elements, irrespective of their order. So yet another possibility is:
CollectionAssert.AreEquivalent(new[] {"joe", "fred", "jenny"}, people.Select(p => p.Name).ToList());
Note that CollectionAssert.AreEquivalent() is a little picky with regard to the types it accepts (even though the signature takes IEnumerable). I usually wrap it in another method that calls ToList() on both parameters before invoking CollectionAssert.AreEquivalent().