|"These parts are running off of an HP Pavillion a1020n."|
Brand name cases often use one piece female wiring connectors on certain headers that are compatible with being used on the possible mboards the brand name case can have in it, but they're not necessarily compatible with being used on the headers on other mboards, and there is often no markings on the female connectors indicating which wire is for what, and no info about that on the brand name web site, other than there may be info about that in the specs for the original mboard used in the brand name model.
At the very least, the wiring connector for the power switch must be connected to the right pins on the mboard. If you're not certain about whether other connectors are compatible, connect only the wiring to the power switch pins to see if the mboard will boot.
Or, simply, briefly, carefully short the pins for the power switch on the mboard to see if the mboard will boot.
You can buy a generic case locally very cheaply that has no power supply, and you wouldn't have that problem, except, possibly, if the connectors for USB headers are one piece they may or may not be compatible with the USB headers on the mboard . Cases with individual connectors for the USB port wiring will work with any mboard.
MSI G31TM-P21 LGA 775 Intel G31
Intel Pentium E6700 Wolfdale 3.2GHz 2MB L2 Cache LGA 775 65W Dual-Core Desktop Processor BX80571E6700
What specific ram module brand and part numbers are you using in the mboard ?
If you chose the wrong ram, that alone can cause no boot, no beeps.
Assuming you're using ram this mboard model supports....
The cpu you chose is supported, but ONLY if the mboard has the latest bios version.
Motherboards sometimes sit somewhere for months after they were made, so when you bought the mboard often is not relative to the date of the bios version the mboard has.
Support page MSI G31TM-P21
Cpu support page
Pentium Dual-Core Wolfdale SLGUF E6700 266 3.20 2M N/A R0 etc.
minimum bios version 7529v46.zip
Your cpu is in part number BX80571E6700.
I'm assuming that has the SLGUF spec.
Version 4.6 Update Date 2010-04-19
Version 4.5 Update Date 2010-01-12
Version 4.4 Update Date 2009-11-25
Version 4.3 Update Date 2009-10-21
Version 4.2 Update Date 2009-08-14
Version 4.1 Update Date 2009-07-10
Version 4.0 Update Date 2009-06-11
If the mboard was made before Version 4.6 was used on the mboard model, you, or a technician at a computer shop, probably need to install a cpu the older bios version on the mboard supports, and flash the bios to the latest version, then you or he can install the cpu you chose and it should be recognized fine. .