105k said:...Are you guys resting your back or holding it straight and tight. ...
Lot's of good advice above, but in addition to the flexibility and fit questions there's also some subtle riding 'technique' in terms of how you sit on the bike and ride and it typically takes some conscious thought for beginners.
Basically your lower back will flatten out in a comfortable way that doesn't require tense muscles or much core strength to maintain if you also rotate your entire pelvic girdle forward in the saddle. You can think of this as pelvic rotation or sticking your butt out a bit but one way or the other you'll comfortably lose that break in the lower back and with it a lot of discomfort if you can teach yourself to ride rocked forward in the saddle instead of straight upright with an associated 'break' in your lower back.
I remember when this stuff was explicitly taught to newer riders and have heard old school stories of coaches putting straight thin boards up the back of young cyclist's jersey's as a reminder to flatten out their backs which more or less forces them to rock their pelvic girdles forward (moving the actual hip joints up and back).
Time trial bike fit is an extreme version of this with your 'taint' pressed hard into the nose of the saddle but on the road bike it's more neutral and relaxed.
Anyway, just like relaxing your elbows and riding with a slight soft elbow bend usually takes some conscious thought for beginners (along with a good bike fit) so does learning to ride most of the time with the pelvic rocked forward which leads to a straight back. If you do it right (and your bike is fitted for this riding position) it doesn't take much if any back strength to ride comfortably with a flat back.
Look at some photos of good riders riding the drops and brake hoods (folks tend to ride more upright on the bar tops) and notice the lack of a 'break' in the lower back and think about why it is you can often tell a serious rider just by how they sit on the bike. Then practice and try to find that comfortable place then frequently remind yourself to ride that way while out on the road till it becomes second nature.
Good luck,
-Dave