They are both right in different contexts . But the emphassis is different . 放书在桌子上 emphasis what are you put on desk (it is a book). But 把书放在桌子上 emphasis where is you put the book . Hope my explanation will be little helpful .

I've heard people say both. Which one to use? Depends on the context.

你怎么老是放书在我的桌子上？

你怎么老是把书放在我的桌子上？

Both of them sound correct. But you're right. You will hear people use the second form more often.

放書在桌子上 sounds like direction translation from English, put books on the table. I personally will not say it is wrong, but it is certainly not that natural. In other words, native speakers do really put it this way. But I am pretty sure that if you say this, people can still get what you mean. There is a sentence structure called "Ba" structure in Chinese and it is commonly used. If you would like to sound more native, study this sentence structure. Of course, it is a bit of complicated though. Good luck.

放is verb,if verb is the first word in a sentence ,it is an order.it is not polite.

it s 把书放在桌子上