Actually, they probably will work, not great, and not the best solution for the long run, but they probably will.
I just obtained a bike with "V" brakes (AKA linear pull) and I put drop bars with normal-pull brake levers on it for the interim set up. Literally, this weekend. I rode it for the first time today and the I got the brakes set up to work OK, but.....
1 - With mine adjusted as well as I could get them, pretty well cut new cable housings (no excessive loops or friction), they do almost bottom out, but not quite. The front is no problem the rear because of cable flex and stretch is worse. I had to set the pad -rim clearance very close. My wheels are nice and true (decent road wheels), so I can get them close enough. This makes it difficult to disengage the brake to remove the wheel.
2 - the brakes seem to be pretty grabby. Because of the short throw of the levers, they don't really grab until near the end of the pull, but then because of the leverage of the V brakes, they really grab, so be careful.
Even though I made them work, my advice is to simply buy some long-pull brake levers for your drop bars. Tektro makes some (RL 520), as does Cane Creek (I think they're called Drop V's). I think you can get into them for $25 - 40 ro so.
My plan is to install some "Travel Agents" instead of buying new levers. They convert the cable pull of normal levers to the longer pull needed for the V brakes. It's not cheaper - it is actually at least as- or more expensive than new levers (I think mine $30 or $35). But, my eventual plan is to use my Shimano brifters on this bike, so I'll need the Travel Agents at that point.
So, the short story is that mine work, but the brakes have to be adjusted very short and they're touchy because of the mismatch. Unless you're really strapped for cash, find some $30 long pull levers.