I see nothing wrong with eating the same whole grain every day if you enjoy it. Many of us eat oatmeal every day, or brown rice, and even whole wheat bread. Like with wheat breads, make sure the label says "whole" rye.
Now, finding an
oil free rye bread is a problem. I have never found one. In fact, the only oil-free breads I *have* found commercially are the frozen Food for Life brand Ezekial and the packaged Dave's Killer Breads, only the green and yellow label ones, though. But we're also supposed to be buying packaged products that are made without salt, and if they do have salt, the per serving sodium level on the label is less than or equal to the calories. The only one I ever found is the Ezekial Lower Sodium one. Some people have said they can get an oil-free whole wheat packaged lavash at Whole Foods, but I've never seen it, and they never mention if the sodium level is acceptable. Bread products usually have more salt than potato chips, so read those labels!
Oh, wait a minute! Just before I pressed the Submit button I remembered that Ann Esselstyn (Dr Esselstyn's wife) uses rye bread in all her videos where she makes sandwiches! She uses Mestermacher rye bread:
https://www.mestemacher-gmbh.com/produc ... /#tab-id-2I then remembered I have half a loaf of this stuff in my freezer! These loaves are about 8 inches long (this is the bread's width) and 3 or so inches wide. It's very thinly sliced - each slice is about 1/4 of an inch thick. The bread is hard and dry and pretty much flavorless, according to not just me but my husband, too. Ann Esselstyn usually suggests triple toasting it so it's even harder and dryer, and her daughter Jane compares it to a roof shingle. But Ann ignores her husband's (and Jeff Novick's) guideline about the sodium level. Each slice of this bread is 180 calories but 240 mg sodium.
If you like thin, dry slices, but want to ignore the sodium guideline, the Mestermacher is the way to go. But I have never seen a rye bread that meets both the no oil and the lower sodium guidelines made commercially. If you have a bakery where you can order a no-salt added, oil free rye bread, or want to make your own, then you're all set to go.
My own opinion? If you have access to an oil-free rye bread but the sodium level is a bit high, go right ahead and buy it, just be careful with the rest of your food throughout the day so your daily sodium total is under 2,000 mg. It's more important the bread be oil (and other fats, like lecithin) free. I would
not buy a bread that had any fats in it, because that is Dr. McDougall cardinal rule that he shouts from the rooftops.