Umbwe Route and the Western Breach. Isolated, and has a few easy class 3 sections for interest.
The only time we saw other people--except for maybe a total of ten who were descending this more direct way because of altitude sickness--was at a couple of the camp sites. These were sites where other popular trails, like the Machame and Lemosho Glades, crossed ours. Otherwise we had the mountain to ourselves, including the summit itself.
We did descend a popular route, and saw plenty of people on that. However we camped only one night on the descent.
Kili report 
