That's a great tip from Denver in the post above - here's my experience with it:
First, looking at the hole is a drill I've done for years, especially when going to another course - I've found it's one of the fastest ways to get enough feedback to adjust to different speeds than what you usually play on.
I switched to a belly putter a few years ago - first an Odyssey two-ball, then a Frankly Frog, and then a Heavy Putter, all of which I liked for various speeds on greens. Out of frustration one day, I tried using the Heavy while looking at the hole for a complete round - and rolled putts in from everywhere. I didn't have the confidence in hitting the ball in the sweet spot with other putters, especially with a standard length putter - but using belly length to anchor the pendulum, plus the fact that hitting a Heavy Putter off center results in almost the exact same distance as on-center, seemed to work for me. I've been mostly using "look at the hole" for a couple of seasons, and it's held up well for me (4 handicap). About the only time I don't look at the hole is on putts inside 8 feet or so, when I can see the hole with peripheral vision.
Anyway, great drill that Denver recommended, give it a try - and if you've got a putter with a big sweet spot, you may just like it as much as I do!