So, it’s my understanding that in Japan it’s considered to be polite to say “itadakimasu” before eating. From what I’ve read, it seems that the meaning of this phrase derives from Buddhist and Shinto religious concepts, however, which gives rise to this question: how would the typical Japanese person respond to a foreigner (who might be a Christian, Muslim, or a member of some similar monotheistic faith) who refused to say it (as politely as possible) and cited religious reasons to do so?
Outside of a temple, or perhaps dinner with the Emperor, "itadakimasu" has zero religious connotations. A reasonable translation would be "thanks for the food" said to no one in particular. The after-dinner phrase is "gochisosama".
No one in Japan will expect you to know the details, so you can just sit quietly and let the moment pass. Display expert use of chopsticks and your dinner companions will not notice your silence.
However, flat-out refusing to conform to local customs after you have been informed of them, for any reason, won't win you any friends. Japan is the land of "do it this way and like it" all wrapped up in a pleasant and polite package. They will call you a rude-ass foreigner after you leave.
Source: 20 years living in Tokyo.
Itadakimasu means, literally, "[I humbly] receive". Yes, the origins of the phrase may be about giving thanks to the gods, but at this point it's about as divorced from that as telling someone "bless you" when they sneeze is about warding off evil spirits.
In any case, a lot of Japanese people don't say it, and they certainly don't expect foreigners to even try. So the odds of you causing offence by not saying it are pretty much zero, and in fact I'd wager that going into some long explanation about why you're not saying it would probably cause more puzzlement and confusion.
Other answers have pointed out that simply not saying it, as an obvious foreigner, will probably go unremarked, as they wouldn't necessarily expect you to have memorized all the customs. (Nor will most Japanese people say anything to your face even if they do find your behavior as a foreigner shocking.) But you specifically asked about refusing to say itadakimasu and saying it was for religious reasons, so let's get into a little more detail about that.
First, you should know that religion in Japan is complex and deeply ingrained, subsumed even, into the culture. There are a number of things average people on the street do without being able to cite any specific reason, especially not a reason we would consider "theological" or doctrinal. It's just... what you do at this place/under these circumstances. Also, there's a prevailing tendency towards syncretism (the idea that all religions are valid or have some truth) rather than the pluralism that's more common in, say, the US ("you do your thing, and I'll do mine.") Shinto coming-of-age ceremonies, Christian weddings, and Buddhist funerals for the same people are all common and not considered to conflict.1
Because of the syncretism, the indirect role of religion in most people's lives, and the general cultural emphasis on harmony and going along with things, it's very difficult to explain why you would refuse to do something for religious reasons. If you pray five times a day, read the Bible, etc., that can be explained by "oh this is how we do things in my religion" (which might be understood as "in my culture"), but the idea that there are specific doctrinal prohibitions against certain things that would be important enough to override the assumption that you'll follow local cultural standards is new to a lot of people.
For example, I did some sightseeing with a Japanese friend once, and we went to a temple in her neighborhood. Even though she knew full well that I was a Christian and in fact working full-time for a Christian organization in Japan, she was still caught off guard when I declined, as politely as I could, to bow, burn incense, or make offerings at the temple. She's Buddhist (a more specific religious identity than most people I talked to held or were able to articulate), but I think it's fair to say that if she went to a Catholic cathedral in Europe, she'd expect to perform whatever ceremonies were customary there, regardless of her personal beliefs.
 I knew a guy who was a "wedding priest" - he had no religious credentials and was in fact an atheist; his only job was to be a suitably foreign-looking white male, dress up, and officiate the Christian-style ceremony. And he said this was pretty much expected, actual Christian ministers being pretty thin on the ground, and not considered offensive by the locals even though he felt weird about it every so often.
P.S. My sense is that, since it mostly conveys gratitude to the people who made the food, saying itadakimasu shouldn't be a problem for any religion I'm aware of - but that's a question for one of the religion stacks. :)
I'm trying to understand the context where you would ever have an opportunity to explain you weren't saying it for religious reasons. Do you really expect them to say "pardon, why did you not say grace with us?" ... and throw open the gauntlet to a religious discussion?
What worries me is that you'd expect or even hope for that.
Yes. They will react very negatively to that.
They would get it as not religious conviction at all, but rather as "I refuse to be humble"... since the statement is all about being humble.
I had a Japanese exchange student stay with me last summer. We say grace at our dinner table where she joins us for the meal. At first she seemed uncomfortable with our prayer format, besides not knowing hardly any of the english words. Eventually she settled on saying "itadakimasu" at the end of our prayer as her own way of saying "amen". After she explained it, we were all satisfied with the intent, and happy to know that she was thankful for the food as well.
Thanks for asking this question. I know my answer is just what I experienced but good to know what others think about it.
Japanese are very tolerant in that respect and have low expectations to foreigners regarding their etiquette. Also they do not consider this phrase to be religious or anyhow important. At worst they would consider you a little weird.
Much more important that you get some other things on the table right like
- respecting seniority when standing up and starting
- using chop sticks according to the essential etiquette
As Peter said, nobody thinks of itadakimasu as having anything to do with religion.
To answer the original question, if explaining that you for religious reasons don't want to say itadakimasu, people would probably just get the impression that you are very serious about your religion, and they would probably worry if you need Halal or Kosher or similar special food.
But, since saying itakadimasu is a well established habit, when eating with foreigners (especially with ones who don't speak Japanese) they may say "Bon Appetit" instead (because of the habit of saying something, and being more considerate to foreigners). This often happens e.g. at business dinners with foreign visitors.