What do I do to avoid noise when using non-USB headphones/mic?

The bugs and driver issues with my G930 has finally pushed me over the edge, and I've now decided to use my old Sibaria headphones with the Antlion ModMic that I bought on Massdrop like a year ago.

Right away I've noticed some issues with both. My headset is constantly buzzing, although quietly, but still annoyingly. Also, testing my ModMic for the first time, when I turn it up high it my voice sounds clear and crisp, but I get the same buzzing/noise that I get in my headphones. When I turn on noise filtering (in Realtek HD Audio Manager) the noise mostly disappears, but my voice gets really muffled and it's much harder to hear what I say.

Can I assume that the built in sound card on my motherboard is to blame for this? What solutions would you recommend to getting the clearest sound through the headphones and microphone with the least amount of noise?

I've searched a bit, and both the Creative Sound Blaster E1 and Creative Sound Blaster X-Fi HD look perfect on paper for what I want (an external sound card/DAC that supports both headphones and a microphone), but I'd be really disappointed is I purchased one only to find out that they produce just as much noise as my crappy integrated sound card.

Any experience and expertise on the subject would he highly appreciated. Also I should probably mention that anything pricier than 200-250$ is probably out of the question, unless it's super amazing :)

An external DAC produces the cleanest sound you can get I think Tek Syndicate even sells one. Before you go running to buy stuff however awesome it may be just go through the checklist quickly. Where did you plug in the jacks? If you are using a panel header try plugging directly into you Mobo case wiring creates noise so front panel jacks often have this problem. Next check your audio drivers if the settings are messed up or if the driver is outdated or corrupted it can cause this to happen as well. If you are plugged in correctly and your drivers are solid then it may just be your mobo's dac in which case you need new hardware.

First I plugged both into my keyboard's headphone and microphone jacks which was pretty awful, at least double the noise that I got from plugging directly into the motherboard.

As you suggested I just updated my Realtek driver and to my surprise the static background noise in my headphones got a lot less noticeable. I can still hear it if there is no sound in my headphones, but just barely. I can also still hear a constant noise in the background when I record my voice, but it's a lot softer and less annoying now. Thanks a lot for that suggestion.

If I still decide that I want an external sound card to get the clearest sound possible, what would you recommend? (It needs to have inputs for both headphones and a mic, and preferably work under Linux)

constant noise is very typical of mobo audio cards since they rarely have proper power isolation / filtering and typically suffer from poor grounding.

An external DAC will clean up your headphone signal. An external capture card would be great for the mike too. Or... just a two-in one like the creative products.

It's not cheap but its really well made and supports the Tek, other than that you have a few choices for actually decent internal options most sound cards will work out well as far as sound cards that work with linux I really don't know but I do know you can get USB based sound chipsets with 3.5mm jacks that work fairly well and they would be your cheapest option.

I had the same problem and assumed it was my sound card. Turned out it was because my plug socket wasn't grounded and only uses 2 pins. I manually ran a wire from the 3rd pin on my power cord (that comes out of the PSU) out the window into the earth, and the noise went away.

That's a good point I'll have to add check ground onto my audio trouble shooting list, although with normally power supplies I don't think that will happen often without actual damage to the plug ends.

As you said an external DAC or audio card is needed and than, to clean up the power, you can use a current stabilizer with electro-mechanical switches (even just with electronic switches is good). Also not using power strips and being sure to not have any radio frequency device on near plugs surely reduces return frequencies or noise into the system. In the end you can also put some ferrite on the power cable of the DAC.