No.
(Hmmm... Stop there or elaborate a little?)
People have been eating grains for thousands of years. But, before that, people thrived for hundreds of thousands of years without grains. And, there are (and were) many traditional societies whose diets are (and were) completely grain free. So, it would seem that, no, a healthy diet does not NEED to contain grains.
But what about nutrients? Aren't grains a wonderful source of vitamins and minerals? Not really. On a per-calorie basis, when you compare grains to an equal serving of non-starchy veggies, the vegetables are a much better source of nutrients. Check out this chart from Jonathan Bailor's blog. Even vitamin enriched whole wheat flour pales in comparison to veggies.
But what about fiber? Although whole grain products have twice as much fiber as, say, a doughnut or Wonder Bread, they don't measure up to non-starchy veggies. A serving of veggies with equivalent calories has seven times the fiber of whole grains. So if you want fiber, load up on veggies (with butter, of course)!
And, if you like skeptical musings, you might enjoy this post, entitled 'Fun with Fiber: The Real Scoop' from Mark Sisson's blog, questioning the very notion that fiber is important in a healthy diet.
Whole grains. Traditionally, that meant the whole grain. As in, the whole kernel. Think: pearled barley, wheat berries, steel cut oats, farro. Grains were soaked, boiled, fermented (like beer), and sprouted. These are the whole grains that have been consumed for thousands of years. Modern eaters tend to turn to whole grain flour. Let's look at the difference, from the perspective of the glycemic index (GI) -- a measure of how quickly those grains turn into glucose in our blood:
White Bread 73 GI
Whole Wheat Bread 71 GI
Coco Pops 77 GI
Grape Nuts 75 GI
Special K 69 GI
VERSUS
Wheat Berries 30 GI
Pearled Barley 28 GI
As you can see, when whole grains are pulverized into flour for breads or cereals, they lose all of that 'whole grain goodness', at least from a blood sugar perspective, and perform very much like white flour. But who wants to eat wheat berries for breakfast? (Maybe porridge? That's vintage.) The reality is that our love affair with grains is almost entirely a love affair with highly processed food - the breads, cereals, crackers, chips, pastas, and pizza crusts made with flour.
So why do we keep hearing so much about “heart-healthy whole grains”? Observational studies. That’s right. Almost all of the science that looks at the benefits of whole grains is based on weak associations (and imprecise food frequency questionnaires). With any epidemiological study, the “healthy user bias” creeps in… after all, who eats whole grains other than health-minded people who have many healthy habits that improve their outcomes? So if you read a headline about whole grains, check to see if the reported result is an actual experimental result or just an unreliable observational association.
Bottom line -- whole grains are a great source of CALORIES. And whole grain flour is a great source of the kind of calories that spike your blood sugar. So, if you need more of that, go for it!