The "AD/BC" way of counting dates that you want dates to 525 AD by Dionysius Exiguus. Prior to that, people kept time in "regnal" time, meaning that they would count the Xth year of the reign of Y. Since the Old Testament was pretty much complete by 400BC (nearly a milennia earlier) and the New Testament by 95 AD, it would be every bit as odd to see an "AD/BC" date as would to see a reference to Miley Cyrus in the text.
The Bible has many, many "regnal" dates, perhaps most famously in Luke:
And it came to pass in those days, that there went out a decree from Caesar Augustus that all the world should be taxed. (And this taxing was first made when Cyrenius was governor of Syria.)
1 & 2 Kings does a lot of this as well:
16:23 In the thirty and first year of Asa king of Judah began Omri to reign over Israel, twelve years: six years reigned he in Tirzah.
Even the prophet Haggai gave us specific dates:
In the second year of Darius the king, in the sixth month, in the first day of the month, came the word of the Lord by Haggai the prophet unto Zerubbabel the son of Shealtiel, governor of Judah, and to Joshua the son of Josedech, the high priest, saying
These are the dates you should be looking for.
Finally, as a simple thought exercise - remember this. BC dates are before Christ. If you knew that you were living in, say, 587 BC, you'd know that the Messiah would be arriving in, oh, 587 years. Kind of removes the need for a prophetic word of God, don't you think?
Update: What about months?
The Hebrew Calendar has 13 months with different non-Roman names. (Or, more precisely, years in which there are two months of Adar) Even today, many countries use different calendars with different month names. Here in Esther, we read:
In the first month (that is, the month of Nisan), in the twelfth year of King Ahasuerus’ reign, pur (that is, the lot) was cast before Haman in order to determine a day and a month. It turned out to be the twelfth month (that is, the month of Adar).
Remember, the Romans didn't even bother to name two of their months, and most of them were ordinal (Sept=7, Oct=8, Nov=9, Dec=10). Months just aren't that important to an agrarian society. They care about seasons - planting, harvesting, starving. Dates are of concern to bureaucrats and historians - and history to a Greek person would be totally different than to a modern (read Herodutus!) Asking "Why no dates" really is like asking a farmer why he doesn't wear a watch, or asking a fish why he doesn't ride a bike.