I'm convinced that line colour can, in some situations, make a difference. I don't know whether John Goddard and Brian Clarke's book 'The Trout and the Fly' was published in the US, but it's well worth getting your hands on a copy, IMO. They have a section on fly line colour, illustrated with some underwater shots. It mainly deals with floating lines, btw, but on the whole sinking lines tend to be dull coloured anyway.
In brief, Goddard & Clarke's conclusion is that while the line is in the fish's 'window' (ie the cone of water immediately above them in which anything is viewed against a background of sky) the colour of line is more or less immaterial, as they will all appear dark. Of course, it might be said that your line should, as far as possible, not enter this zone at all - only the leader.
But outside the 'window' is the area they call the 'mirror', where light hits the water at too acute an angle to pass through the water's surface, and is reflected off it. In this area, which extends considerably further than the 'window', any fly line will be seen against a background of a reflection of the river bed. In this area, a green or brown line is significantly less visible than a white or brightly coloured one.
If you fish places where the water's surface tends to be rippled, I suspect there's sufficient disturbance to the 'mirror' that a line will appear broken up and not make so much impact. On the other hand, if you fish very flat water or are covering very spooky fish, I think line colour can make a difference. Dana has mentioned New Zealand as one place where this view is widely held, and there are other examples. In the World Championships in Aragon last year, for example, one British competitor noticed that his catch rate increased when he switched to a dark grey floater, but that that the most successful teams were using silk lines (which tend to be dull-coloured). They reckoned this gave them 3-4 casts at ultra-spooky trout, where a plastic line generally allowed only one shot.
When a similar topic occurred recently on another board, the most sensible advice given, IMO, was to use as dull a line as you can manage, without compromising other aspects of your fishing. Thus, if you find a bright line helps your casting, or allows you to detect delicate takes at distance, then by all means use one. But if you use a dull line you certainly won't harm your chances of success, and may well actually improve them.