The official Library of Congress transliteration (thank God for it; do you know how many ways there were to transliterate "Tchaikovsky" before?) for the Russian version is Dmitri. I agree with jmbejdl, though. There's nothing more annoying than being told you're misspelling (or even mispronouncing) your own name.
I've also had a priest insist on using another form of my name, but this was an all-convert "we're Americans and proud of it--so proud we'll wipe all vestiges of "ethnicity" from our parish" group, so the priest actually insisted on anglicizing my middle name. He even changed it on paperwork. I'm all like, "Dude, you can't change my name," and he's like, "But we're Americans" and I'm like, "But it's my name. It's on my birth certificate. It's what my grandma calls me." Very stupid.