Using only analogous colors works better for fine art, for example, Van Gogh's Sunflowers above, for the applied arts such as Graphic Design and Web Design if you are going to use analogous colors it's probably better to include at least one accent color.
Analogous Colors
Analogous colors are colors that are adjacent to each other on the color wheel.
With analogous colors you can create harmony but you have to be careful or your design can become monotonous.