It's a frequently asked question in English study forums in China (recently here http://www.douban.com/group/topic/31374821/). How do you say "第几" in English? When A asks B, "这是你第几次来纽约?" (literally, "This is which time you come to New York?"), B may answer "第二次" ("The second time"). A more natural English question may be, "How many times have you come to New York?", "Twice", or "How many times did you come to New York before?", "Only once (before)".
The awkward "which time" is a literal equivalent of "第几次" as in "Which time is it you come to New York?" The English "time" is one word for both the time you use a clock to keep track of and the ordinal count of repetition of you doing something, which is a measure word. Other languages may use two words for these meanings (时间, Zeit vs. 次, Mal). The reason "which time" sounds unnatural may be related to this particular polysemy (one word having multiple meanings) of English "time".
Here are more challenging ones, "第几个", "第几件", and "第几本", as in "老师要我们读John XYZ的ABC系列的所有三本书，你在读第几本?" ("The teacher wants us to read John XYZ's all three books in the ABC series. Which book are you on|reading now?") But "Which book are you on?" is not a good translation because the answer may well be "I'm reading his Book Title". The question actually demands the answer "I'm reading his first|second|third one". English "which" properly matches "哪一(本)". It does not specifically ask the ordinal number as the Chinese "第几(本)". The fact that these Chinese question words are more challenging is probably because "个" and "本" have no measure word equivalents in English, and although "件" may be "piece", "Which piece ...?" does not specifically demand an answer of the ordinal number in the series.
A little follow-up. A Chinese reader says "So English is deficient?" My answer is that every language may have stronger expressive power than another in one case, but less in another. In this case, Chinese wins. In the case of subjunctive mood, ...
“第几名”是特例，因为英文的"What place was he in?"的答案基本只能是"First|Second|Third|... place". 即是说，没有除用序数（ordinal number）以外的方法能回答的。而"Which book are you reading now?"的答案就既可以是“I'm reading Harry Potter”也可以是"I'm reading the third book".
我重述一下我的问题。汉语“这是你第几次来纽约？”字面上对应英语Which time is this you come to New York?但这句英语不地道，没有人说Which time，而只是说How many times did you come before? 又如，汉语“你在读老师规定读的第几本书？”，回答只能用序数词，如“第三本”，但英语Which (book) of all the books the teacher wants us to read are you reading now?既可用序数词回答（I'm reading the 3rd book），也可用其他方式（I'm reading Gone with the Wind 或 I'm reading the one by Jack London）。