The rainy season in the Andes and the Amazon region lasts approximately from December to March. In the mountains, this period is definitely the wettest, as rain may fall almost everyday for several hours in a row. Mornings are usually dry, so it is recommended to use the early hours for moving around. Temperatures are not very low during the day and night, and the air stays relatively warm as well. In the Amazon region, the rainy season doesn't really differ from the dry one, as rain is a frequent phenomenon year round and temperatures are high, which creates an almost constant steam effect.
In fact, rain shouldn't scare you, otherwise you will miss a chance to experience some great things in Peru. Surprisingly, some tourists prefer trekking to Machu Picchu via the Salkantay trek or Inca Trail during the rainy season when you don't get stuck with crowds of people on the way. Also, some mountains are good for ice climbing only at the end of the rainy season. In December you can watch or even join the fighting festival Takanakuy, or visit the vivid carnival during Fiesta de la Virgen de la Candelaria at Puno in February. Remember to try fresh lucuma during its harvest period and try a favorite Peruvian cocktail Pisco Sour on its national day in February.