Banská Bystrica is a city in Slovakia.
Northeast towards Brezno, Poprad 66
Catch bus 33 towards Senica and get off near the end of the line when you see the Shell gas station along the road. Use this same bus stop to hitch from. Don't miss your stop, or you'll end up in a sketchy area of town!
Another option is to search for the train station, continue walking to the direction of Brezno, and you will meet a bridge and a Kaufland. Next of it there is a quite good place for cars to stop and pick you up!
North towards Ružomberok 59
Catch bus no. 43 and get off at the at the end of the city, bus stop "Novy Svet II", walk 50 meters along the road and there you will find a good place for cars to stop. If you are not lucky, walk another 300 meters along the road, there is a gas station where you can ask for a lift.
South towards Zvolen, Bratislava, Budapest R1
Catch any bus going to the Tesco on Zvolenska Cesta, and stand under the bridge close to the Radvan railway station. If you're going to Budapest, you can take a train to Zvolen for about 1EU (as of October 2012) and walk from the train station to the spot in the intersection of 59 and 66 (it's marked on the map).