I think that most of the better known Bibles done today are pretty accurate... or at least pretty faithful to what at least some manuscripts (MSS) said.* The only problem I can think of with the Douay Rheims is that, if it is (?) based on the Latin Vulgate by Jerome, it will have some passages and tendencies that you won't find in texts used by easterners. Jerome had many discussions with Jews in his life, and eventually was convinced by their arguments, and this colored his scholarship somewhat (I recall even the Catholic Encylcopedia article on Jerome at New Advent admitting this). Also, some Orthodox actually accept more books in the Scriptural canon than what the Catholics accepted at Trent (namely, 3rd Maccabees, Ps. 151, Prayer of Manasseh, 1 Esdras--I think Catholics call 1st Esdras "3rd Esdras" since they already have a book by that name in their canon--and a few other passages here and there, such as at the end of Job). Obviously if you want to read these (other) books, you'd have to get a different Bible.
The Orthodox New Testament mentioned already is an ok Orthodox translation. The patristics and notes are helpful, but I would disagree with the claim that the translation is excellent: I think it's awkward and chaotic in it's rendering and punctuation. Still, it is probably the best Bible in English put out be an Orthodox group. If you wanted to read all the scriptural books many Orthodox accept, including the ones named above, your best bet would probably be The New Oxford Annotated Bible With the Apocrypha. Or you could do what I do, and just grab the "other" books from online sources and use a Bible that you already own 
I tend to use the KJV and NKJV, but that's just me.
Justin
*Though there are obvious, and easily discernable, exceptions. Usually they'll come right out and tell you when they are doing a loose paraphrase and not trying to follow the MSS closely.
