As I have no doubt that Nir's answer will work spending a little time implementing it, I was able to do what I wanted a little bit more elegant with this code:
Private Sub myWindow_LocationChanged(ByVal sender As Object, ByVal e As System.EventArgs) Handles Me.LocationChanged
Dim primaryBounds As System.Drawing.Rectangle = Windows.Forms.Screen.PrimaryScreen.Bounds
Dim windowBounds As System.Drawing.Rectangle = New System.Drawing.Rectangle(CInt(Me.Left), CInt(Me.Top), CInt(Me.Width), CInt(Me.Height))
If (windowBounds.Left < 0) Then
windowBounds = New System.Drawing.Rectangle(0, windowBounds.Top, windowBounds.Width, windowBounds.Height)
ElseIf (windowBounds.Right > primaryBounds.Right) Then
windowBounds = New System.Drawing.Rectangle(primaryBounds.Right - windowBounds.Width, windowBounds.Top, windowBounds.Width, windowBounds.Height)
End If
If (windowBounds.Top < 0) Then
windowBounds = New System.Drawing.Rectangle(windowBounds.Left, 0, windowBounds.Width, windowBounds.Height)
ElseIf (windowBounds.Bottom > primaryBounds.Bottom) Then
windowBounds = New System.Drawing.Rectangle(windowBounds.Left, primaryBounds.Bottom - windowBounds.Height, windowBounds.Width, windowBounds.Height)
End If
Me.Left = windowBounds.Left
Me.Top = windowBounds.Top
End Sub
This made the window being dragged stay within the primary screen (whole window), but you could easily change the bounds to whatever values you needed.