+ 1 - When I started riding (mountain biking) I used to get a lot of neck pain, hands falling asleep and even lower-back pain. After a couple of months of riding 2-3 per week, my body got used to it. All these issues diminished considerably, to the point that it was never a problem. I did make many adjustments along the way.

Try the following:
* Wear comfortable ergonomic gloves
* Advise your bike shop that you're experiencing discomfort on rides over 10 mins. Ask them to please go over with you on seat and handlebar adjustments to see if they detect anything wrong. Small adjustments help!
* On your rides, stretch your neck, shoulder and arms during short stops

Try the above for about a month. Give your body time to adjust. If the problem persists:

* Ask your bike shop to see if you could try a different stem. Playing around with the stem's length and inclination helps a lot in this type of circumstances
* Consider a new handlebar - It may be that your body simply is not conditioned for drop bars. Believe me, I've tried and tried and I just don't find them comfortable whatsoever... and I'm young and physically in good shape and have been riding bikes for 7 years now continually. On my commuter bike, I switched to northroad handlebars. I know lots of people who also enjoy the moustache handlebars. Should you go that route, buy them on the internet and have your bike shop install them. Keep in mind that you may have to change the brake levers, too.

Others have mentioned joining yoga or pilates classes. That's a huge plus if you're able to do that since you use a lot of your core muscles when cycling.