You can use my interparc tool, as found on the file exchange. It is designed to solve exactly this problem, in 2 or more dimensions.
xyz = interparc(1000,x,y,z,'spline');
xyz will be a 1000x3 array, where each row is an interpolated point on the curve.
Of course, you can do the interpolation in other ways. If you want to do the work in MATLAB yourself in much the same way as you tried to do it:
t = [0;cumsum(sqrt(diff(x).^2+diff(y).^2+diff(z).^2))];
t = t/t(end);
ti = linspace(0,1,1000);
xx = spline(t,x,ti);
yy = spline(t,y,ti);
zz = spline(t,z,ti);
The above solution, which uses a cumulative piecewise linear arclength as the parameter for interpolation, deals more properly with points that are not uniformly spaced.