Towards Ayacucho, Cuzco
Go towards Plaza Heroes, then continue straight for 3 gas stations; the last one is by the road going to the left, you can start hitching here. There are almost no private cars and the road is full of taxis. The trucks pass there around 11-13.
Altough the chances are good that someone will pick you up, the road going down to Ayacucho is crazy difficult. It´s 3m wide in most parts, muddy (in December 2012 the greater part was not paved; they are working on that slowly, however), bordering with a canyon, there´s traffic, kids playing on the road, farm animals, rivers - that grow significantly in the rainy season - and plenty of memorials for those who died on the way (in the most interesting parts of the road all of the above together). Going 240km might take you more than 24hrs with one car and you´ll damn yourself for chosing this trail - but yeah, if you have time and nerves of steel - it´s crazy beautiful. Good luck!