1. Fill bottle/vase half full with warm water.
2. Drop in 2 Alka Seltzer tablets. These are the kind I had:
I had to break the tablets into pieces to fit into the opening.
3. Cover the opening and give the bottle/vase a good shake.
*Use caution. My shirt looked like this after shaking:
According to the internet, you just rinse and voila, it's miraculously clean. Since these bottles have been sitting on the side of Highway P for Godonlyknowshowlong, I let them soak for about 30 minutes. After changing my shirt, I went back to my bottles and repeated steps 1-3. Round 1 had loosened a little bit of the gunk, but nothing noticeable. After round 2, I decided to use a q-tip to wipe out what I could reach. This helped, but still no miracle was happening. I squirted dishwashing soap and hot water in each bottle and let that sit for another 30 minutes. Looks like I'll be buying a bottle brush. :)
This may work better on a freshly used vase, and maybe it was the type of Alka Seltzer that I used. I also found a pin that said it could be used to clean your toilet. Guess what I'll be trying next! And no, I will not be sharing those before pics.