Colca Canyon [1] and other stunning areas near Arequipa [2], like the Majes Valley, are one big adventure playground. There is a huge range of options, including Class III–V rafting, mountain biking, trekking, climbing, and good agencies to make it happen.
Before you go, remember the altitude: Acclimatize in Arequipa.
Links:
[1] http://www.moon.com/destinations/peru/lake-titicaca-and-canyon-country/colca-canyon
[2] http://www.moon.com/destinations/peru/lake-titicaca-and-canyon-country/arequipa
[3] http://www.moon.com/destinations/peru/lake-titicaca-and-canyon-country/arequipa/recreation/climbing
[4] http://www.moon.com/destinations/peru/lake-titicaca-and-canyon-country/arequipa/recreation/trekking
[5] http://www.moon.com/destinations/peru/lake-titicaca-and-canyon-country/arequipa/recreation/rafting-and-kayaking
[6] http://www.moon.com/destinations/peru/lake-titicaca-and-canyon-country/arequipa/recreation/biking
[7] http://www.moon.com/destinations/peru/lake-titicaca-and-canyon-country/arequipa/recreation/tour-guides-and-agencies