Its OK not to use H1 tags. On page optimization techniques and page structure matter much less to SEO now than they did even two years ago. You still need to use keywords and synonyms in the text of your pages and the title tag is the place in the page that matter the most. Beyond that, using H1, bold, and link anchor text may help a very small amount if done in moderation. Used too much, with too many occurrences of the keywords you wish to rank for, then your site can suffer an over-optimization penalty.
Three years ago to rank for a keyword the formula was
- Make a page featuring that keyword
- Put the keyword at the beginning of the title, in the h1, use it several places in the page maybe with a bold
- Give the page lots of internal pagerank by linking to it from your site extensively
- Start building external links to that page that use the keyword of the anchor text.
Today that is a formula for disaster. Here is what you need to do today:
- Build a section of your site (at least several pages) that cover all aspects of that keyword.
- Use the keyword and its synonyms in a natural way. Put the keyword in the title, but not always at the beginning, use other related phrases in the titles too.
- Link the pages in that section together and link to the section of your site internally, don't use the keyword as anchor text too much.
- Do user testing and make sure that your site satisfies people looking for that keyword. (Keep in mind satisfied depends on whether they are looking for info, ready to buy, trying to navigate somewhere, etc.)
- Build links to the pages externally. Anchor text should be 50% domain name, 20% "click here" variations, 10% page title, 2% keyword, 18% other. Don't pay or spam for links.
As you can see, Google has made "doing SEO" much harder. They are now trusting on-page signals much less than they used to. They are paying more attention to user reaction to your pages. They are penalizing much more often for keyword stuffing and spammy link building.