I recently changed some things on my bike, resulting in a more aggressive position (new stem has no rise & there's a pretty big drop from seat to bars). For the first 2-3 weeks I was getting a soreness in my neck/shoulders and in my lower back. But as I've adjusted to the new position (and probably built a little more strength), the pain has gone away, even on long rides.

I also find that it's useful on longer rides to periodically check in to see how relaxed I am. I'll open up my hands and "play piano" on the drops, take a deep breath, and make sure I'm not tensing anywhere from my upper back/shoulders/arms down to my hands. I find that since I've been doing this, I'm more relaxed all the time and actually need to do it less.

Core strength really helps too -- if I'm starting to feel tired or tight, I'll get down really low in the drops, so my elbows are at a 90 deg. angle, and grasp the handlebar very lightly kind of high in the drop, so I'm actually NOT carrying a lot of weight with my arms/hands, but rather supporting it with my legs and core.