Before
Don't give up on your running plan when you check the forecast and look outside - you won't melt in the rain! Also, if you're actually expecting rain, you can prepare for it and kit yourself out according to the forecast. In light rain, you'll need a wind- and water-repellent jacket, while in heavy rain, you'll be better off wearing a waterproof jacket and trousers. If you're running with your trail bag, you could also bring your waterproof outfit (that packs into tight balls) to be prepared for a change in the weather. In any case, you can't affect the weather conditions on race day so you might as well endure them during training and make the most of it to test your gear before the big day.
Having said that, make sure that your phone, MP3 and money are well protected to avoid any inconvenience.