Featured Links are now called Featured Promotions. These instructions are for Retina version 4.0 and up. If you are on an older/pre-sectioned version, please update your theme.
Since the rollout of the Shopify Sections platform, adding more Featured Promotions and controlling the number of links per row is easier than ever. To setup your Featured Promotions, go to Admin > Online Store > Themes > Customize > Add Section. Here, select and add 'Featured promotions' to enable a default section of 3 featured links in a row. Then, using the settings depicted here, you can add/remove Promotion blocks, and take control over the number of promotions that appear in a row: