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Here's a LINQ version of Fredrik's solution. Note that Enumerable.Range yields an IEnumerable<int> so you have to convert to chars first. Cast<char> would have worked in 3.5SP0 I believe, but as of 3.5SP1 you have to do a "proper" conversion:
var chars = Enumerable.Range(0, char.MaxValue+1)
.Select(i => (char) i)
.Where(c => !char.IsControl(c))
.ToArray();
I've created the result as an array as that's what the question asked for - it's not necessarily the best idea though. It depends on the use case.
Note that this also doesn't consider full Unicode characters, only those in the basic multilingual plane. I don't know what it returns for high/low surrogates, but it's worth at least knowing that a single char doesn't really let you represent everything :(