As far as I know, there is one route on the mountain which has been done, and it involves crossing the south col at about 5500m and rapping down to the SW glacier and setting up high camp. Summitting then involves crossing the glacier to the north ridge. I had seen this route from the north and west, and it looked similar to climbs we'd been doing. Also, we figured we needed to get up and down and back to Namche in about five days, and our Sherpa Gomba, who'd been on the mountain, said that route would take ten days and involve much schlepping of gear. So we thought we'd try a direct route around a hanging glacier and straight up the face.
We started from Macchermo around 6am, and picked our way up the talus through the clouds for several hours, aiming for "base camp." In fact, because Kyajo was first opened only a couple of years ago, there is neither a climber's trail nor an obvious place for a camp, but nevertheless we were surprised to find an Australian team -- most hospitable -- at a base camp who had already ferried their gear over the col and were ready to try the regular route.
The weather was breaking up and we could see our proposed route, which skirted below the hanging glacier to gain the rock on the left side. When I mentioned our plan, the Australians' sirdar looked very serious and suggested we might want to watch to mountain for a while. This was a new approach to route-planning to me, but we soon saw his point: more or less regularly every half an hour the hanging glacier was dropping a Himalaya-size load of ice about 200m down the face, basically scouring the route. Since we figured it would take us an hour+ to cross the impact zone, that was the end of that plan. Since we didn't have the time and equipment for an extended siege, we ended up descending.
If I do it again I'll approach from the SW side, and I'll keep in mind that global warming is having a huge effect in the Khumbu now. Glaciers I walked up directly in years past are now riddled with cravasses; bergschrunds are all opening up in a major way, and walls I remember as rime-covered ice problems are now bare rock.