You can hitch on Silvan Blvd, right after the crossing with Elazig Blvd. There are a few buses going there from the city center, ask the locals.
Buses E9, A2, A3
Option 1: From Ofis district (and from other parts of town as well), you can take a minibus (dolmuş) to the İlçe Otogarı (bus station for short distance destination). Then just cross over the railway and you'll find yourself on the D950 leading south to Mardin. Tapiroblu never waited more than 20min here at this spot. You can also get to this spot by walking from the train station on Mehmet Akif Ersoy boulevard, it's like 15 minutes.
Option 2: Another option if you are in Diyarbakır's old town (Sur) is to just get out from Mardikapı (Mardin gate, the southern gate) and try to get a ride south, on Old Mardin Road, beyond the Ongözlü köprü until it connects to D950. But option 1 is better.
Accommodation and Sleep
There's an old hippie living in Diyarbakır who owns a cozy place called Cafe Mahya, close to the Ofis Mosque. He is very active on couchsurfing, hosting hundreds of people every year. Update 13/09/2014 : locals said that this cafe Mahya is now closed