Prostration was common throughout the Biblical period and remained daily practice for many Jews into the medieval period, especially in Muslim countries. Maimonides in Mishneh Torah, Hilchot Tefila 5:13-14 discusses prostration in the context of the post-Amidah suplications/tachanun. He seems to be writing descriptively (this is what Jews do) rather than prescriptively (this is what one should/must do), and says that some have the practice of 'kida' (sitting on ones knees and then folding one's body over them and stretching one's arms straight out on the floor) while others do 'hishtachavaya' (lying completely flat with both arms & legs extended straight, a 'superman' type pose). It remained the practice for tachanun among Yeminte Jews until relatively recently, and continues to be practiced by some individuals here & there. In Ashkenazi communities it is only preformed during a few key moments during the High Holidays.
The only flat-out prohibition on prostration is that it cannot be done on stone floors outside the Temple in Jerusalem. Not very many places where Jews pray really have stone floors, but I suspect this prohibition is behind the common behaviour in Ashkenazi communities when prostrating during the High Holidays to put a towel or tallis bag beneath one's head.
Nevertheless, because it is not common in any large Jewish communities today, there are issues with doing it in public. The Talmud records that during his prayer Rabbi Akiva could be left in one corner of the room and later found in the opposite corner because of his many bows & prostrations. It goes on to note that he would not do so in public so as to not inconvenience the community. Some authorities, for example Rav Moshe Feinstein, have interpreted this story to mean that one should not do things in shul that are seen as strange or outlandish and will cause a commotion or distract people.
My personal practice when I'm alone or in my yeshiva is to prostrate during tachanun, but I don't do so in shul for the above mentioned reasons. I do think it's a shame that the practice has fallen into disuse, though.