Flat UI brings about a dramatic change in the way color is used in web design. Since the designs are no longer skeuomorphic, designers are more reliant on color and color contrast to relay information on the screen.
Flat UI tends to use more saturated, bright colors instead of grey, white, or black. You can use many different shades, as long as the tone and depth of the colors match. Often, the more simple color palettes are used, but the primary objective is to use colors that help convey the content as opposed to using color to mimic a 3D everyday object. The key is to go for simplicity.
There are a number of online tools to help you create a flat UI color palette. You can take a look at http://flatuicolors.com for a good sample of color codes. You can also visit http://www.colorhexa.com/ for a good color-matching tool.