Took me a little while to find a solution for this problem.  But as usual, posting a simple question on the XNA MSDN forums prompted tportz to provide me with a great chunk of code that worked perfectly (so thanks belong to that poster).

Here is the code in a nutshell:

public float UpdateThumbStickAngle(Vector2 stick)
{
    float angle = 0.0f;
    stick.Normalize();
    if ((stick.X >= 0 || stick.X <= 0) || (stick.Y >= 0 || stick.Y <= 0))
    {
        angle = (float)Math.Acos(stick.Y);
        if (stick.X < 0.0f)
            angle = -angle;
    }
    return angle;
}

Fairly simple chunk of code, but the math part in the middle is pretty much over my head.  Thanks again to tportz for providing this simple solution that would have driven me nuts otherwise (and was starting to, which prompted my original post to the MSDN forums).

In other news, I will be posting my "Geometry Wars Clone" some day this week, I just need to clean up and better comment the code so it makes sense to others ;)

Download source code that contains an example of a triangle sprite that you can freely rotate around in a full circle using the Left ThumbStick of an Xbox 360 controller.

RotateSpriteThumbsticks.zip - 10.50kb