Is the H1 tag important for SEO?

I spent some time finding hacks to be able to specify H1 tags for my WP site, but am now thinking I really don’t want that H1 text to appear on my pages. It’s seems “old fashioned” or something like that to always have a “heading” at the top of the page. And I’ve read that you shouldn’t create H1 tags with no corresponding text on the page. So for now I’ve just left the standard WP H1 tags in place which of course is the site/blog title, the same on every page which is not good either.

Any thoughts about this?