I am hesitant about the best way of cutting the blocks in half as cleanly as possible and without removing too much material.
Let me present a slightly different tack here.
Looking at the first link you posted, it looks very much like the blocks are an exact fit on the tubes:
This is actually pretty simple to achieve. First, cut two blocks that are each half of a cube. Then, clamp the blocks together and drill the hole for the tube such that half of the hole is on half of each piece. This is best done on a drill press.
Like @rob states, you might want the kerf taken out anyway so that you get some clamping action on the blocks. If that's the case, then either of the answers posted by @rob or @keshlam will work perfectly fine.
Edit #1: If you want the clamping action while still following my original advise, shim between the blocks with some thick paper or card stock or thin shims of wood. It will essentially be the same as taking the kerf out with a saw.