Absolutely. Your website design has alot to do with how search engines rank your website, to a point. Search engines cant see your perfectly layed out website design or that perfect shade of blue your using on the outside, but it does the everything going on behind the scenes on the inside.
Search engines crawl your website periodically to check and see if there are any updates to your site or structure, which can influence rank. The structure of your website coding or programming has everything to do with your SEO. Search engines have certain things they look for in the HTML of websites, they want to see a well structured, up-to-date standards and techniques. You must make sure your website’s HTML is using the right title tags, alt attributes, lists, paragraphs, content, internal linking structure, etc. Search engines also cant read images so it is important not to use images as important headlines and use proper header tags in there place. Proper website structure is the main portion of on site search engine optimization.
Search engines cant read flash websites. Period. If you care about your search engine ranking positions you should steer away from a fully based flash website. Flash can be a great tool to add some flare to some sites but does nothing but hurt your rankings. Flash should be used for small elements such as headers or video displays only, even the use of a small flash navigation is no good as it destroys a bit of your internal linking structure.