Here's my run-about mountain bike... I just converted it last weekend (still haven't put the rear brakes on in this photo, it was after a test-ride to the store).



This is a 2001 bike so it still uses the stem-quill system. I was going to get a Quill-to-A-Head converter to use a modern universal bar clamp which you likely will need for modern bars.
Instead (in my case) I found that by prying open the stem clamp slightly (carefully) and sliding in aluminium drop-bars from an old Raleigh 10-speed I was able to achieve the right fit.
Any sort of this kind of conversion generally means replacing v-brakes with center pull cantilever's - IMHO. Check Sheldon Brown's website for a description as to why (mechanical advantage differences) - http://www.sheldonbrown.com/canti-trad.html
The basic summary is you need to match the right brake levers with your brakes - with cantilever's you can match them with standard road levers, none of my bikes have disc's but they may need special considerations as well. There are road levers for v-brakes or you could use a travel-agent device but I found it's just simpler to put in cantilevers and be done with it.

Anyways, it's a blast to ride this bike and the multiple hand positions/dropped riding style suites me much better. Straight bars always cause discomfort in my hands/wrist after an hour or so on the bike.
I used it to haul some stuff today (one of it's many functions) with the saddle bags on the rack and it still handled great.

For shifters I clamped a set of Suntour downtube shifters (from the same Raleigh) on ahead of the original cable bosses (so it can't slide back) and routed the cables as normal. This wouldn't work if you have a bike with top-tube cable routing. If you are using indexed shifters you might have issues trying to get that working, however the deraileurs will still work fine if you switch to friction-shifters of any sort.

I spent a lot of time on Google and rummaging my parts bin as well as pestering the several LBS in the area for odd parts and pricing... It's best to have a plan before starting as these projects can escalate in cost rapidly if you don't.
For example, it's fine to put cantilever's on a bike that had v-brakes but you're going to need to add center-pull cable hangers... I added the down-tube shifters but because of the difference in tubing width I needed a slightly longer bolt to attach them.

As already mentioned, you're likely going to need your bars set higher, and with a shorter stem than you would with straight bars.

Here is what the bike looked like last year with 'normal' bars, shortly after being used for a bike move (after the trailer was unhooked naturally):