Boy does it matter!!!!
 

Working all week at Farm Bureau Tour Event with a player. Working on full swing and "checking" on the putting. It is amazing how painful a good ball striking round and a poor putting round can be. In fact, I would go to say that poor putting makes golf the least amount of fun possible.

With that said, make sure all of this is not just talk, you need to be walking the walk. Don't go a day without working on your hinges, tracing the line, knowing the amount of accumulator #3 you use, or using a stationary head. Even if you check them only for a minute. It is so vital to "lowering" those scoring averages and winning those matches.

Doing the math on a great ball striker it is easy to see that putting is AT LEAST 40% of the score. Not to mention the "momentum" that is carried and lost during a tournament or round by the putter. Taking it a step further it is always the last "shot" of the hole or round or tournament and we all know the excitement of the last chapter in the book.

Watching the majors we all kinda feel the pressure of the players over that last putt to win a major. You get that "hair raising" feeling just watching, IMAGINE WHAT THEY FEEL LIKE. So considering the amount of "pressure" that putting naturally places on the player is the final "nail in the coffin" when it comes to a reason your putting stroke mechanics should be magnificant.

Amen....I had a poor putting year... yes, YEAR.. about a month ago I started watching (in a mirror without a ball)my stroke. I couldn't keep my head stationary. Somehow over time I had developed movement that I wasn't able to pick up unless I looked down into a mirror. Then it was obvious.... It took about 4 days of working on it before it disappeared. Now I don't let a day go by without looking down into the mirror and monitoring the path... I used to do this while hitting balls, but as per your suggestion I started doing it without a ball. It has helped tremendously. I do anywhere from 30 to 50 very deliberate strokes and then try and work gaze into it also. I'll repeat that twice sometimes a third time if my back holds out. Last two rounds...27 and 29 putts... nothing spectacular there either, just making 4 ft and 5 ft. putts that I was consistently missing.

A better way?
 

It's probably too late for us now, but ever think that Sam Snead had the right idea with side-saddling? The eyes are on the line as we face the target line, the motion is simpler, and if you hang the putter from your right shoulder, vertical hinging never deviates the clubface from the plane line.
There is always one guy at every club who used to be a yipper and now makes everything side-saddling.
I wish I had learned it as a kid. The long putter is pretty sweet, though.

Like tossing a ball underhanded ;)