When you Google your name, chances are that your Twitter account will be one of the first results that shows up. Google has designed results from Twitter to come up in top results and your tweets will be searchable if you haven't set your Twitter profile to private. Now, of course, the argument is that if you don't want to show up on Google, you need to set your Twitter account to private, but if your account is public and you still don't want to show up in Google, Twitter wrote a post on its support blog to help you get started.
Don't let Google clip your wings (or steal your tweets)
The first three steps include your information. Twitter recommends that you change your name in your profile settings as well as change your username and (surprise surprise), protect your tweets. When you change your username, make your new handle something that does not resemble your actual name. However, if you begin protecting your profile when you previously had a public profile, the tweets you sent when you were public will still be indexed in a public search engine including twitter.com/search. What this means is because Google caches search results, old information will still be posted. Until Google updates the new information and indexes your current status, links to the profile or updates posted prior to removal/protection remain online.
Once you protect your tweets, any old links that show up in Google's search results will take the searcher to a page saying, “This page doesn't exist”. If Google still hasn't updated your information, you can choose to remove your content from Google yourself (this can also work if your results still show up on Google but upon clicking on them, you are taken to the error page). Copy the URL of the content you would like removed from the site which is the Twitter page of the exact tweet you want removed. You then head over to this page. You paste the link of the Twitter page you want removed and then submit your request. For more information on removing content from Google's search index, click here. In order to submit a content removal request, you might need to sign in to your Google account or create one if you don't have one.