I am trying to power my pi through the 5V pin on the header, through an lm317, for a clean 5V output. However, when I connect the 5V to the board, the pi constantly reboots itself. I am assuming this is due to low voltage, and I do have other loads in the circuit, but I believe I am well within the 1.5A max that the lm317 provides. Can someone explain to me why this is happening?
Rather than trying to guess why your Pi (which Pi model) doesn't work can I suggest you just give up.
The LM317 is a 40 year old chip (I didn't know you can still get them) and I haven't used one for over 30 years. They were never a particularly good regulator, and you would only use one if you really needed a variable voltage. They have a quite high Vin-Vout differential, making them even more inefficient than fixed regulators.
You MIGHT be able to get 1.5A out of the regulator, but this would require a substantial heatsink.
You would certainly need adequate decoupling to ensure stability.
A voltmeter or better yet a scope on the 5V (as previously stated) will answer if the 5V is stable enough to handle the load. The LM317 may show unloaded stable 5V but if input to the LM317 is not high enough voltage or current the output will not be stable. Input and output caps shown are standard suggested values. Not shown in circuit but a good add on for ripple rejection is a 1N4002 across the 220ohm resistor