The details of Monet's cathedral painting show the colors he avoided using simple black for shadows.Photo: © Andrew Yee (CC)
Think about how much is truly black in nature. Shadows are not simply black nor a darker version of the color of the object. They contain the complementary color of the object.
Take, for example, the shadow on a yellow object. If you mix black and yellow, you get an unattractive olive green. Instead of using this for the shadow, use a deep purple. Purple being the complementary color of yellow, both will look more vibrant. If you can't figure out what colors are in the shadows, simplify what you're looking at by placing your hand or a piece of white paper next to the bit you're having trouble with, then look again.
Or if you can't see yourself working without black, then consider mixing up a chromatic black rather than using a straight-from-the-tube black. It also has the advantage not 'killing' a color it's mixed with to the same extent.