A lot depends on WHERE and HOW you would be hunting. A grizzly might not weigh more than 300-500 lbs in many places. Along the coast of SE Alaska they get much larger.
There is a big difference between killing power of a .270 with a thin jacketed 140 gr. bullet and a .375 mag. throwing a tough 250 gr. bullet.
Bullet technology has changed all the old rules about what we used to think would work and what would not. There are a lot of tough 270 and .30 bullets that will penetrate plenty. A 270 or 30-06 with proper bullets will work just fine in a hunting situation. What they may not do is stop an attack quickly if a bear decides to do so. If that happens you will likely have a guide standing beside you with something along the lines of a 375.
If I were going to Alaska I'd take my 30-06 or 300 WSM and load it with 200-220gr Nosler Partitions or 180 gr Barnes TTSX bullets. Both of those have proven that they will shoot all the way through a grizzly from most any angle. A 270 loaded with comparable 150-160 gr bullets would do almost as well probably.
The bigger bores would give you a little extra insurance in case things go bad, but I doubt it will matter.