I was having a lot of problems with my weight in putting. I could hit it straight, and read the breaks decently, but I've always been somewhat gun-shy when it came to actually hitting the ball. I didn't want to end up hitting it six feet past the hole and then have a difficult second putt, so I ended up wimping out of a lot of putts and leaving it six feet short, which didn't help anything.
I decided to try putting with my eyes closed to see if I could get better results. I went to the practice green with two balls and just kept putting to each of the holes. I would take a practice swing with my eyes closed, step up to the ball, line myself up, close my eyes, and putt. I realized that my body is a WAY better putter than my mind. I was getting the ball to stop withing a couple of feet of the hole consistently. I manged to get good enough results that I decided to do the same thing on the course.
I've played three rounds since then and my putting has been getting better every round. I still have the odd three-putt, but most of the holes are two-putts or better.
I don't always make good contact with my eyes closed so my goal is to keep letting my body decide the weight of the by taking practice swings with my eyes closed, but then take the real putt with my eyes open to make sure I hit it flush.
I actually really enjoy putting now, rather than being afraid of it.