Silk Cloth

Farming tip: If all you want is silk cloth, and don't want to keep stopping to dump the junk, fill all your bag slots with one piece of cloth. Bags will be full for everything but the cloth and cash.
To make room for an AH sale item, just combine two slots into one. Auto-loot and leave the junk behind. This is especially useful when almost full and you have to dump grays too often.

I decided to try out five different places that were listed on here and other sites to farm silk. Here's what I personally got from each. I soloed ALL of these on my level 62 Disc Priest with crappy gear, and had no issues anywhere. In order of best to worst, they are:

Blackhoof Village, Dustwallow Marsh:
This spot is THE best. Blackhoof Village is very tiny -- takes maybe 30 seconds to walk through end to end, but the respawn rate on mobs is insanely fast (30-60 seconds usually, sometimes faster). For reference, one loop around the village took me about 3 minutes, killing and looting included. I constantly saw them spawning right in front of me, and sometimes my cooldowns couldn't even keep up. However, they're lowish level mobs (32 I think) and not very aggressive, so it's a really easy area to farm. I farmed here for about 2 hours and I got:
420 Silk Cloth
30 Mageweave Cloth
1 blue
9 greens
1 recipe
If you need Silk Cloth, this is the place to go. Even if you don't stay very long, you'll leave with a ton of mats. If you go here, you definitely won't need to go anywhere else for Silk Cloth, and you'll have plenty to put on the AH. Equip large bags before going though, or prepare to drop things while there. I'd advise you clean out your bags first and store anything you don't absolutely need in the bank if you're looking to farm a lot.

Uldaman, Badlands:
Second best that I've found. I spent about 10-15 minutes farming on my way in, and then about an hour inside. I hadn't done this one in forever, so I got a bit lost. If you know the instance, it'll probably be quicker. Some of the boss fights took awhile, but they're not hard. If you're higher level, you can probably run through and one-shot most things. The drop rates were very good here, in my opinion. An hour and 15 minutes gave me:
129 Silk Cloth
55 Mageweave Cloth
6 blues
1 greens
If you're looking to get a lot of silk and some mageweave, and are willing to spend some time, this is a good spot. Most mobs were dropping either Silk Cloth or Mageweave Cloth, and I got quite a few weapons and armor to DE along the way.

Scarlet Monastery, Tirisfal Glades:
I first ran through the Forlorn Cloister area of SM. It took me about 16 minutes total, from start to finish. After that, I headed to the Library Wing. For about 34 minutes worth of farming, my total was:
93 Silk Cloth
39 Wool
7 blues
2 greens
IMO, it's an okay drop rate. Just over half an hour will get you almost 5 stacks of Silk Cloth alone. Not the first place I'd go, but it's definitely not the worst. If you don't need a ton of Silk Cloth, this might be a good place; it's pretty quick to get through.

Gnomeregan, Dun Morogh:
I got about 20 minutes into this and then gave up. The drop rates were bad, at least in my experience. In 20 minutes of farming, I had:
19 Silk Cloth
1 Wool
1 blue
2 greens
Keep in mind, this is less than half of what I got in Scarlet Monastery in slightly more time. Not worth it, and I wouldn't recommend going there to farm Silk Cloth.

Gahrron's Withering, Western Plaguelands:
I found this place on a WoW leveling guide site. It said to farm the Haunting Visions and Wailing Deaths. For ~20 minutes worth of farming, I ended up with almost nothing. I got:
4 Silk Cloth
A few silver & copper
This is NOT a good place to go whatsoever. The drop rates are extremely low and there's not many mobs. Note that the same guide had the Cult-Sworn Pugilists and Cult-Sworn Warlocks as mobs to farm (they are north of Gahrron's Withering), but I didn't bother going after how dismal my results were in the first area. This may have been a good area previously, I don't know; if so, it got seriously nerfed.

For reference, these were done between 1:30pm and 8pm EST, on the US-Shadowmoon server (on an Alliance toon). Hope this helps those of you looking for good spots to farm! :)

Well, I decided to go ahead and bring up a tailor, so that meant searching out the metric ton of silk I'd need.

I started trying out some of the places listed here, but frankly they weren't doing well enough for me. The popular spots of New Thalanaar and Blackhoof Village seemed promising at first, but the respawn rate just couldn't keep up with me at 90. If I'm waiting on respawns, I'm not getting cloth, and if I'm not getting cloth, I'm wasting time.

Instead, I took a trip to Eastern Plaguelands and started laying waste to Death's Step Miscreants and Death's Step Putrifiers. The drop rate may not be the greatest {or maybe it is ;) } but there was always at least 1 or more packs respawning no matter how fast I was taking them out.

30 minutes of constant killing netted me 240 Silk, 1 blue, a slew of greens and the occasional piece of mageweave. Whoohoo!