OP is using a datasheet diagram for the regulator.
But he/she has no idea on what a PSU actually consists of. Instead we are given the runaround.

Why do you people even bother ?
That's why I ask the question direct.

If you want my help, show me what u got. There is no other way.

 

I'm using 4 diodes, 1N4004.

Would the small capacitor after the rectified output damage the chip? The chip no longer functions, even in the simple breadboard circuits with a filtered 9v DC supply. I think that's the part I'm having the most trouble understanding. What's causing the chip to be damaged?

U can blow all the chip that u can buy and nothing will help unless u tell us what u are doing.

Take a picture and post it for god's sake.

 

If you have a 34v DC output with a 1000 uF filter capacitor, this is a perfectly fine input for the LM317 (CHECK).

The LM317 has a weird pinout. CHECK and DOUBLE-CHECK. If the part is sitting in front of you, right side up, pins pointing at you: adjust, out, in <--CHECK

Be careful with where your 240 ohm feedback goes. Once again, CHECK the pinout.
Be careful where the 6k POTENTIOMETER used as a RHEOSTAT goes.

Carefully, hook the input voltage V+ to the INPUT pin. Tie the Vgnd from the DC input source to the bottom of the RHEOSTAT and measure the output voltage on the OUTPUT PIN and Vgnd.

YOU SHOULD be able to twist the knob and source an amp with a good heat sink into any load you want.

FOR GOOD TRANSIENT RESPONSE, STABILITY, and NOISE IMMUNITY, add the capacitors as shown in the DATASHEET.

IF YOU WANT TO POWER INDUCTIVE LOADS: Place a diode, anode-to-cathode, from output to input. This protects the LM317 from electromagnetic kick-back from the inductive load (relay, motor, etc.)

Hopefully this helps.........

P.S. - I ASSUME you're using the LM317, TO-220 package, from the LM317T part number you referenced.

 

Yep check the wiring to the LM317 pinout. It's the number one reason for LM317 circuits not working.

And don't forget to add the very important cap from the adjust pin to ground (that people forget to put in their schematics)!

 

I've got this running reliably now. Thanks for all the help and hints!

Thanks for the explanations PaulEE. Also, I'll include schematics in the future. It was simply my ignorance which caused me to exclude it. It seemed like such a simple circuit...

I've also installed LTSpice so I could create diagram of the actual circuit I built (Before and After). I initially included the sample circuit from the datasheet since it is such a simple chip. That's why it was so frustrating that I didn't understand how it was getting damaged.

I still don't understand why the chip would be damaged by the filter capacitor being too small. I understand that the voltage as jumping between close to 0 volts and 34 volts, but since those voltages are within the operating range, why would the chip become damaged?

The pinout did throw me initially, but I solved that long before I posted here for advice.

 

...
I still don't understand why the chip would be damaged by the filter capacitor being too small. I understand that the voltage as jumping between close to 0 volts and 34 volts, but since those voltages are within the operating range, why would the chip become damaged?
...

It probably wouldn't get damaged from input ripple, provided the voltage was less than the max safe voltage. The LM317 regulator is designed to tolerate input ripple very well.

I think people were communicating that it is both a standard practice and a very good practice to use a decent sized filter cap after the bridge rectifier and before the regulator, for any power supply.