For the past 3 years, I've done a guest-lecture in an ancient history class on ancient Roman & Greek religion. In those cultures, there were the established civic cults where everyone would go to the ritual and then the barbecue (comparable to a 4th of July celebration in the US, I suggest), but there was also the personal religion, the one-on-one deals, the ex-voto offerings and side-bets with God. These almost always operated on a quid pro quo basis: if God/deity does X, then the suppliant will do Y. If X doesn't happen, then the two parties go their separate ways and a polytheistic worshiper will try another deity.
This the model for Hannah asking for a child and promising it to God (1 Samuel 1.11), or any number of examples from ancient Greece & Rome. The students in these classes are always understandably dismissive of ancient religious belief, but they grow quiet when I remind them that, if the prayers and requests didn't work, there wouldn't be statues, dedications, and inscriptions telling us that the people believed that they did. The sheer number of ex-voto offerings suggests that these people believed that their prayers had been answered. Curse-tablets asking for horrible and violent actions have been uncovered spanning about a thousand years and huge geography. Ancient religion was inherently practical - if one god didn't fulfill your needs, you went and found another god. (note the 1st commandment to the Hebrews suggests that there are lots of options out there, but that they should have no other God before Jehovah; this is different from declaring that only one God exists). So, if these things didn't work in some way, why did people keep doing them for hundreds and hundreds of years? There was no belief (like today's) of "pay your tithing and then God will bless you temporally." It was "If God blesses me temporally, I'll pay some tithing."
The pragmatic nature of these transactions means that the dedicators considered their prayers and requests to have been fulfilled. Who am I to discredit their loyalty and integrity in fulfilling the human end of the vow, when today's religions rely much more heavily on things like faith and patience and confidence in a final judgment? Most religions today expect their followers to obey God, submit offerings, and to meekly submit to whatever God wills. Ancient peoples expected to pay only for services rendered.
Even if I don't believe in the gods of other times and other places, I sense a reverence or sacredness in places where people recognized their gods, whether it's Chartres Cathedral, a temple in ancient Greece, or a Native American vision-quest site. Maybe the hallowed nature of the ground stems solely from the human effort expended on that spot, and the human hopes and reverence towards something outside of themselves, but it's still there. At least to me.