Is there a better way of cleaning a sieve, other than stabbing at it with a brush, in a manner such as this ?
I can't help but think there must be....
You could get yourself a specialized sieve brush - the things are meant for mining sieves but cooking sieves are generally the same size (0.5mm - 1.0mm, the latter being the no. 16 that the first link is talking about). I haven't seen any of these sitting in home kitchens - could be that they aren't effective on cooking sieves, but more likely, cooks just don't know or care that they exist.
For the truly dedicated, you could get yourself a studded sieve cleaner or the similar triangle kind with brushes. These are meant for cleaning really fine sieves, as in flour mill equipment, so whether or not it would be useful to you depends on what kind (mesh density) of sieve you have.
Or, if you could loads of money to blow, you could go all out and get yourself an ultrasonic sieve cleaner. Of course, we're now talking about products that don't even have a list price, so if you have to ask, you can't afford it.
Here's perhaps the most interesting option for those of you at home, though: I noticed companies out there selling rubber balls as sieve cleaners, so I was curious whether or not one could just use an ordinary rubber ball. I happened to have one of those big squishy stress-reliever balls sitting around and gave it a shot on my 8" sieve/strainer that had some sediment stuck on it - and guess what, it works! It's not perfect, it didn't get every last grain out, but I only had to rub the inside a couple of times with the ball to get out the vast majority of "dirt". So there's your "home remedy" for cooks on the cheap.
I've never found this to be a big problem. If I have something dry in there, I give it a few whacks on the divider in my sink, then run it under hot water and scrub with a sponge for a minute. If there is something really stuck on, I might soak it in hot water first. Or if I have room, I'll throw them in the dishwasher. What do you have on your sieve that is giving you particular trouble?