You can clean both dynamic and condenser microphones using the same method.
Start by carefully unscrewing or removing the microphone's grille. Then wipe it using a solution of lukewarm water and a mild detergent such as dishwashing liquid and a soft cloth. Gently rub the grille with a toothbrush to remove the dirt. You can also use a gentle disinfectant if any residue is left even after using the dishwashing liquid. Finally, let the freshly cleaned grille air dry without getting a drop of water in your microphone as it can damage it.