Typically it would have no or minumum effect on what search engines think of the website. They mostly focus on content, not looks.
Your design should be there to encourage visitors to do what you want them to do. It can influnce the usability of the site, how visitors perceive you, and conversion rates.
No, change in theme would not affect the SEO and Traffic diretcly.
But, there is an angle to it.
If with new theme, the user experience goes down and bounce rate of your website increases, then it could affect the SEO of your website. As mentioned by Tony, search engines look for content and usability, doesn't matter what theme you use.
Hope it helps :)