Sadly, bins are pretty useful when it comes to reasonably storing things without quantum stockpiling. Stone blocks are one particular example of an item that really benefit from bins, but can't themselves be efficiently routed to various construction sites without minecarts or vast amounts of dwarfpower (which isn't really very efficient, come to think of it).
Dwarves will put bins in wheelbarrows when they're moving them (and the items inside) from stockpile to stockpile. If you want to use particular materials, you'll need to pull the blocks out of any general stockpiles into a single-material block stockpile, and then you can use wheelbarrows to move the blocks from there to the construction site. If any material is fine, you can skip the specific stockpile.
Alternatively, you could use wheelbarrows to haul the desired stone to the construction site and make the blocks there. I believe, based on a failed hauling setup, that if you order a workshop to give to a stockpile that doesn't take the produced items you can stop those produced items going into a stockpile that would normally take them.
I tend to limit wheelbarrows to very particular items like stone because trying to get bins to behave with wheelbarrows has been a nightmare for me.
For example, a dwarf wants to put one brick into a stockpile. The stockpile has a bin with 4 blocks and room for a fifth. The dwarf will grab the bin full of bricks by hand, carry it allll the way over to the single brick, and THEN run back and get the wheelbarrow. But... my haulers usually aren't very strong and a bin full of 4 bricks is pretty heavy, so it takes for-fing-ever.
As an update, I THOUGHT the dump-to-takefrom stockpile solution was working until I noticed that the bins-stealing-items-from-binless-stockpiles bug was resulting in a huge stream of dwarves running the bricks back DOWN the tunnel to put the bricks in bins back in the stockpile they just came from, even though both are set to only take from links, and the outdoor stockpile (the final destination for the bricks) is the only set to take from the stockpile the bricks are presently in.
A) Links Only Workshop output pile -> B) Minecart which dumps to single tile stockpile set only to give to -> C) Links Only Outdoor brick stockpile.
I think at least 20% of my bricks end up back down in the outpile pile because dwarves are SO AGGRESSIVE about binning items. OCD little jerks.
The only workaround I can think of is to only use bins on ONE stockpile for that type of item, ever, if you want to make absolutely sure your items don't get shuffled about by dwarves with BIN MADNESS.