Are you taking breaks by riding on the tops? You may need to do that to relieve the muscles that hold your head up when you're on the hoods or drops.
Are you trying to go too far too fast? If so, you have to realize that riding a road bike setup as a road bike requires some gradual adaptation, because with the more forward position, there is more work for the muscles that hold your head up. People should not expect to buy a road bike, hop on it and ride long distances right off the bat.
Is your helmet too low on your forehead? This forces you to lift your head up more in order to see where you're going. It's very hard on the neck. Similarly, if you happen to wear glasses, these always tend to fall too low on the nose, and like the too low helmet, it forces you to lift your head more than you would have to if you were not wearing glasses.
LBS/pro fit or no fit, are your handlebars very low? They don't have to be at saddle level, but there isn't much to be gained by having them much lower than an inch below the top of the saddle.
But all in all, it does take some fitness. I get neck pain after about 10 miles whenever I get on the bike after a long layoff.