The best recommendation is to take all necessary precautions for yourself, wildlife and the fauna.
What measures make sense depends on the area you are visiting. For example, in more remote parts of the east coast (Maine, New-Brunswick, etc.) you will only find black bears and they tend to be pretty shy. In these areas, many people rely on hanging their food, as the bears usually prefer to avoid people, and good trees are plentiful.
In more heavily-trafficked areas such as National Parks, bear encounters are more likely, and the bears tend to be less fearful of humans and savvier about how to get at your food. Some southern portions of the Appalachian Trail now require bear cans, either year round or at least during peak hiking season. The same is true in popular areas of the Adirondacks. In the western portion of North America, where you are more likely to encounter grizzlies, carrying a bear can is a very good idea. Grizzlies are larger and more aggressive than black bears, and in parks like Yosemite and Yellowstone the bears have food bags pretty well figured out. They recognize bear cans as well and will often knock them around in an effort to get at the good stuff inside. Note that in many Parks you will be fined if a Ranger catches you backpacking without an approved canister, and in these situations a bear can basically pays for itself!
However you decide to store your food, you may want to consider odor proof bags such as OPSAK or NyloBarrier. By keeping all your food in a sealed odor-proof container (along with any food scraps, wrappers, dirty dishes, and scented articles such as soap or chapstick), you will reduce the likelihood of attracting animals to you camp in the first place. Added bonus: keeps your snacks from getting soggy when it rains all night!
An important consideration with rodents is that if they get to your food, you should be wary of contracting diseases such as Hantavirus which can be fatal.
You may also want to consider a food bag made of Cuben Fiber:
On several occasions I've seen mice check out my Blast food bag and eventually give up. I let one persistent mouse chew on it for the entire night- by morning he had done some minor damage but did not get into my food.
as per Joe Valesko. My experience was similar but your millage may vary.