Just an alternative because it looks quite nice :) - NOTE: Henks answer is the quickest of these.
input.ToCharArray()
.Where(c => !Char.IsWhiteSpace(c))
.Select(c => c.ToString())
.Aggregate((a, b) => a + b);
Testing 1,000,000 loops on "This is a simple Test"
This method = 1.74 seconds
Regex = 2.58 seconds
new String
(Henks) = 0.82 seconds