I have often with some of my blogs just posted a theme on the site and started blogging but this is no way to grow a site, there are some things you really should check for in a theme before starting to use it. I will try to explain what the 4 most important things are:
Tags
Tags are one of the most important things for SEO. Search engines regard tags as the second highest (in my opinion) thing on your blog for indexing purposes. Tags also increase page views from visitors clicking on relevant tags that has to do with the current article they are reading. If you're new theme does not have tags look for another one if you want to have a lot of search engine traffic.
Categories
Categories are for humans more than search engines, if you dont have categories someone cant easy find what they are looking for and the result is less page views and less traffic ultimately. Categories point out the different sections on the site and makes it easier for the visitor to know where he/she can find what they are looking for.
Monetize
If you blog for a living or plan to make some money from your hard work you really should choose a theme that accomodates adds that are not annoying and fit in your theme and site.
Images
if you have a blog that will be using a lot of photos first check your theme's width. The wider the content area the better looking photos will be. One mistake I have made before was to use a very narrow theme for photos. That makes the photos very small to fit in and also makes it harder for the visitors to see if they really want to click on the photo. If you are going to have a lot of photos choose a wide content are blog.
That's the 4 most important factors according to me when choosing a theme for your Wordpress site.