The search index is a view mode, but it is hidden by default (for some unknown reason).
Goto the Manage Display page for your content type. Expand the Custom Display Settings at the bottom. Tick Search Index and save.
A new tab, Search Index will appear. Click this, and then configure your content type to show / hide the fields you want indexed.
You may need to dump the search index (admin/config/search/settings) and then reindex the site. I can't recall if saving the search index does this automatically, of if I normally form alter that form to do it. The easiest way to do this is with drush:
drush -y search-reindex
drush -y search-index
As a side note, I will be releasing a small module to help debug search issues and check things like this. I am test driving it on a few sites now, and think it will be ready soon.