Most people don't care about a restaurant newsletter; they search you up because they are looking to solve a problem. That problem might be hunger, entertainment, connection with a friend.........
So how do you get them on your list? Give them a sample of what you're offering. Not a discount, a sample.
Once they are on the list, send them a personalized welcome email and a campaign with short emails every few days highlighting how you could solve their problem.
Oh yea, don't bury the sign-up form at the bottom of the home page. Have it the first thing they see when they come to your website.
"Your websites should be 100 percent focused on signing up strangers to give you permission to market to them" Seth Godin
Got 60 seconds? Check out the video.