Cuzco is a city in Peru.
North-east towards Pisac and Sacred Valley
Take combi called "huerto" (1 Sol) until Tambomachay, which is about 7km out of the city. From there is pretty easy to find a ride direction north. A cheaper alternative is just climbing the road that leads to the ruins of Sacsaywaman and searching for a ride already from there.
South towards Arequipa and Puno, East towards Puerto Maldonado
You can take a "Los Leones" bus (S./ 1,20) from either Plaza Limacpampa/Calle Limacpampa Grande or Avenida de la Cultura towards Saylla/Tipón and jump off at any suitable place or gas station ("grifo") as soon as it left the urban area of Cusco. We got off at the Petroperu gas station (-13.58297, -71.809832) in between the villages of Saylla and Tipón. Although not many people stopped here to fill up gas, it was still a good hitchhiking spot with good visibility and cars not going too fast. You could tell your drivers that you are aiming for Urcos as this is where the road splits towards the Southern Andes (Arequipa/Juliaca, Puno) and the Amazonas (Puerto Maldonado/Brazil). As usual in Peru, it might be wise to inquire about money expectations before you get in. Be prepared for little traffic on the route towards Puerto Maldonado!
From the Terminal at Llanlacuyoc and Av. Jorge Ochoa, take a bus to Izcuchaca. It costs S/1.50 as of August 2016 (S/2.00 on weekends). After getting off, keep walking towards the end of town. In August 2018, it took User:atopia slightly more than an hour to find a direct lift to Nasca by standing on the road next to the Primax gas station. As usual in Peru, it might be wise to inquire about money expectations before you get in.