  We report on a new 3D numerical code designed to solve the Einstein equations
for general vacuum spacetimes. This code is based on the standard 3+1 approach
using cartesian coordinates. We discuss the numerical techniques used in
developing this code, and its performance on massively parallel and vector
supercomputers. As a test case, we present evolutions for the first 3D black
hole spacetimes. We identify a number of difficulties in evolving 3D black
holes and suggest approaches to overcome them. We show how special treatment of
the conformal factor can lead to more accurate evolution, and discuss
techniques we developed to handle black hole spacetimes in the absence of
symmetries. Many different slicing conditions are tested, including geodesic,
maximal, and various algebraic conditions on the lapse. With current
resolutions, limited by computer memory sizes, we show that with certain lapse
conditions we can evolve the black hole to about $t=50M$, where $M$ is the
black hole mass. Comparisons are made with results obtained by evolving
spherical initial black hole data sets with a 1D spherically symmetric code. We
also demonstrate that an ``apparent horizon locking shift'' can be used to
prevent the development of large gradients in the metric functions that result
from singularity avoiding time slicings. We compute the mass of the apparent
horizon in these spacetimes, and find that in many cases it can be conserved to
within about 5\% throughout the evolution with our techniques and current
resolution.
