Running cables under carpet. Any great methods?

If you can lift the two locations of entry and exit of the carpet without causing damage. Rent or borrow an electricians fish tape or better yet the fiberglass stiff fish rods to push a string through the pull the cable back through. But make sure it is not in a traffic area as repeated crushing from foot steps will eventually damage the cable and cause a short. If the new location is not crossing a traffic area, e.g. hallway, adjacent room etc., you may consider tucking the cable under the baseboard trim, I have found that a coax can usually fit in that space.

Pulling up the carpet IS a pain in the ass and I would avoid it unless you know someone, (or want to pay a carpet install guy), to lay it back down all nice and neat, tucked properly at the edges/under the baseboard/on the tackstrip. You can really make things look crappy if your "carpet-laying" skills are as bad as mine were. I did it once, made a friggin' mess, had to call the "PROFESSIONAL" to come fix what I had screwed up.
However, depending on the length of your cable run, you might have success with one of those "electricians" fish tapes as Theo suggested. The big issue doing it that way is getting the leading tip of it slid under the carpet without jamming-up and snagging on the padding. You gotta make the cable go inbetween the underside of the carpet and the top surface of the padding. Don't try and get the cable under the padding, it's a disaster and besides, the padding is usually glued down to the subfloor. It'll be a frustrating mess. I ended up doing 2 things to insure success: (1) just spend a few extra dollars and call the carpet guy to lift up, and correctly re-install the carpet. You'll be glad you did. And (2) use good quality cable right off the bat. Don't be skimping, using cheaper shit just to save some cash. If you're anything like me, you'll end up wanting to tear the "bargain" cable out and replace it with another one, (which I did 4 times in a 10 year period), and that's a pain. Depending on whether you plan to run the sub line leval? or speaker leval? you'll want to go with FLAT stuff. I have used Analysis Plus, and Nordost. Both worked well. Both have many different "levals" in their lines, so you wouldn't have to sell your firstborn, or auction off one of your body parts to own a high quality subwoofer cable! And the fact that they are FLAT cables means your carpet will look less stupid, or bumpy and bulged-up where the cable runs. (Things like that matter in my house 'cause my lovely wife can't bear the sight of wires the size of garden hoses running across the floor). As for going wireless? I personally do not like wireless subs, Had one once, seemed to have issues with occasional signal loss and interference. Maybe something was wrong with the transmitter module? But I didn't hang on to it long enough to figure it out. Dumped it less than a month after buying it and went the Wired route, never looked back. Good luck with your installation.

Yeah, fish tape is the easiest. If this isn't flat wire, which I'm suspecting it is not, you may have to make runs in and out of a corner rather than going straight across the room. Don't want to create a mole tunnel in the middle of the room.

There is underlay under the carpet that is stapled to the floor so using a fish tape may be difficult . The last time I went the full route and removed the carpet and underlay and routered out a trough in the plywood next to the wall , this worked out very well , the better half decided it would be a good time to have new carpeting installed , so the project cost a little extra but we were both happy when it was done . I have also ran cable along the corner and had custom baseboard made to cover it .