While there are alternate trails the BRR landcare group are working on rehabilitating the area and creating a proper mountain bike trail. For the moment cyclists are best to stay on the walking trails.
The John Cousins Walking Trail is probably the best section for riding. The surface of the trail is predominantly hard packed dirt though some sections are grass covered and a few corners on the main descent are sandy. The trail is well marked and fairly clear as to where it goes. Be sure to follow this trail in a counter-clockwise direction as this gives you a longer descent and easier climbs.
On the return path you'll be greeted with a sign advertising the Kangaroo Trail, this trail is recent and is marked only by a slashed path through the grass as well as a few posts. You'll not get lost as it is short and stands out from the brush, but it is very rough to ride on. It meets back up with the John Cousins trail about 200 metres past the point you originally departed from.