Straight vinegar or vinegar with a squirt or two of dishwashing liquid will also lay weeds low, but it may take repeated applications to do the job. The more acidic the vinegar, the more effective it will be at controlling weeds, but it also becomes more dangerous for you to handle as the concentration increases.
"Regular" grocery-store vinegar typically has 3 to 5 percent acidity; you may be able to find 10 percent vinegar at a restaurant-supply store or where supplies for pickling are sold. Repeated applications of vinegar will acidify the soil, making it harder for future generations of weeds to get a roothold.
Speaking of scorching, you could also invest in a handheld flame weeder--basically a propane torch with an extended nozzle--that lets you wipe out weeds without any herbicide at all. You can find flame-weeder nozzles that attach to a gas-grill-sized propane tank by means of a long hose, or small models that use a I-pound propane tank you can carry easily. An advantage of a flame weeder is that you can use it in the winter to rid your patio of treacherous icy patches, too.
Another popular organic method of dealing with weeds is boiling water. Organic growers sometimes use steam instead of flame to control weeds, and you could simply wander about your patio with a teakettle, scalding every pesky plant you see. Not every weed will succumb easily to this method--and depending on the size of your patio, this may be as tedious as hand weeding but with the added risk of splashing yourself with boiling water.