The 3 oz TSA limit is a challenge when you want to bring your own laundry detergent for a multi-week trip. I brought 3 oz of "Method 8X" detergent which is highly concentrated. Unfortunately I wasn't very impressed as every time I used it, it left a sticky residue on my hands that was difficult to rinse/wash off. It also released the black dye in a pair of black socks bigtime, which ended up ruining a pair of underpants as the black dye seeped into them (my fault for hanging them too close to each other).
I also brought the little dry-detergent sheets sold in camping stores. I wasn't very impressed with those either, as it was hard to get them to dissolve and suds up a sinkful of even just 4 or 5 garments.
Next trip I'll probably go back to bringing a 3 oz bottle of regular laundry detergent and just take a chance on running out before the trip is over. Hotel shampoo could fill in for the last few washes if necessary. And, one of the hotels we stayed in had a deal where they would wash and machine dry a whole load for 10 euros, which we thought was definitely worth it considering that we were able to go out and enjoy being tourists while the innkeeper dealt with our dirty clothes.
On another laundry-related note, I can't praise Rick's braided clothesline highly enough. Not having to hassle with clothespins is a godsend, and the velcro fastening design is ingenious. It performed spectacularly in a variety of different hotel rooms; I was always able to hang it inside the shower so no dripping occurred in inappropriate places.
I would wash our socks & underwear in the afternoon, after housekeeping had made up the room. By morning the garments were dry enough to be hung in the regular closet and continue airing until fully dry.