I just made some changes to your code to show you a different way of getting a similar result. Basically, just update the ydata for each handle in a for-loop, either by looping over the handles or by doing it in one line (what I've done here). I've added another point to your lines to get rid of the stem plot.
n = 6;
A = rand(n,15);
%create 3d points to be graphed
[x,y,z] = deal([A(:,1),A(:,2),A(:,2)],...
[A(:,3),A(:,5),A(:,5)],...
[zeros(n,1),A(:,4),zeros(n,1)]);
%values to update with later
y_new = A(:,6:end);
%graph
figure;
axis([0,1,0,1,0,1]);hold on
view(3);
grid on;
box on;
h = plot3(x',y',z', '.-',...
'LineWidth', 1, 'MarkerSize', 10);
%if you want to define custom colors, define n-by-3 matrix "colorv", e.g.
%colorv = rand(n,3);
%set(h,{'Color'},mat2cell(colorv,ones(1,n),3))
for i = 1:size(y_new,2)
y_new_c = mat2cell([y(:,1),y_new(:,i),y_new(:,i)],repmat(1,1,n),3);
set(h,{'ydata'},y_new_c)
pause(1)
end
