313 bytes of Ruby
c=gets.split.map{|n|n.to_i}+[0];l=[" "*(c.length*5)]*((c.max-c.min)*2+2)
c.each_index{|i|h=c[i];h==0&&next
(h<0?(h...0):(0...h)).to_a.each{|b|y=c.max*2+1-b*2;x=i*4;s=(x+7)..-1
4.times{|a|l[y-a]=l[y-a][0,x+[0,0,1,3][a]]+['|___|/','| | |','/__ /|','__'
][a]+(l[y-a][[(x+6)..-1,s,s,s][a]]||'')}}};l.each{|e|puts e}
(There is one newline that doesn't count, it's only there because otherwise there would be a very long line in there.)
I used the obvious method of making a 2D array of characters and pasting in the cubes one at a time, bottom to top and left to right.
