Here are a couple things I don't like about using non-English ways of referring to God/Jesus:
1. It comes across as super-arrogant. You know the guy that always has to use Latin expressions when there are perfectly good English ones? He's annoying, right? He comes across like a pompous jerk. Sure, he may not really be a pompous jerk. But you'll have a hard time convincing most people that he isn't one. If this is right, and it is morally acceptable to use the usual English words to refer to Jesus, for instance, then using the Hebrew would be an unnecessary turn-off. That's reason enough not to use it.
2. I can sort of understand why someone would want to use Yahweh, or YHWH, since the Old Testament was written in Hebrew. I certainly don't think it is required. But the New Testament, however, was written in Greek, and the English "Jesus" is just a transliteration of the Greek word "Iesous." So, disparaging a transliteration of the word used in the inspired Word of God to replace it with a word that was not used in the inspired word of God is absurd. What? Paul didn't know the word Y'shua (or whatever)? If he did, then why didn't he use it in the New Testament? The pro-Hebrew-names crowd can't say, "Well, he was writing to a Greek-reading audience." After all, they think they should use Hebrew names when speaking to an English-speaking audience, no? So Paul should have, too. But he didn't. And he was inspired by the Holy Spirit -- which is more than we can say for anyone using Hebrew names today.