If you hold a chunk of bare uranium, your hand won't burn and nothing will glow. The real danger isn't the radiation—it's what happens if you forget to wash your hands afterward.
Despite its reputation as a deadly nuclear material, natural uranium is safe to hold. Its safety profile comes down to its long half-life. Uranium-238, the most common isotope, takes 4.5 billion years for half of a given sample to decay. Because the decay process is so slow, the metal emits very little radiation at any given moment.
The radiation uranium does emit consists almost entirely of alpha particles. Alpha particles are relatively massive and carry a strong electrical charge, making them highly ionizing but poor at penetrating matter. They are easily stopped by a sheet of paper or a few inches of air. Most importantly, alpha particles cannot breach the dead outer layer of human skin (the epidermis). As long as the uranium remains outside your body, the radiation cannot reach your living cells, and your external exposure is practically zero.
The actual threat of touching uranium has everything to do with chemistry. Uranium is a toxic heavy metal, similar in its physiological effects to lead or mercury.
If you handle a piece of raw uranium and then touch your mouth or eat a meal without washing your hands, you risk transferring microscopic uranium dust into your digestive system. Once inside your body, the risk profile changes. With no protective layer of dead skin to block them, the alpha particles are free to bombard living internal tissues. More immediately, the chemical toxicity of the heavy metal targets the renal system, where it can cause kidney damage.
The standard protocol after handling depleted or natural uranium metal is simply to wash your hands thoroughly with soap and water to remove any heavy metal residue.