Short of nuking from orbit, of course. I want to be able to use the walkways this summer. 
Living in Saskatchewan, I’m naturally aware of high grade herbicides that kill pretty much anything unto the 7th generation, but I’ve got the Cub running around and don’t want anything particularly nasty and lingering.
So, is there any product people would recommend that are green-ish in the killing department, but don’t linger?
I pour boiling water on them; they dry up and die in a few days (especially in hot, dry weather), then sweep out vigorously.
Glyphosate is much maligned but highly effective and if you avoid the panic merchants and read something relatively sober about its toxicity it seems to be safe.
Try this from Cornell:
http://pmep.cce.cornell.edu/profiles/extoxnet/dienochlor-glyphosate/glyphosate-ext.html
salt or vinegar would do the job and last longer than boiling water
I agree. I find boiling water very effective for killing weeds.
Salt has a nasty habit of washing out into other areas, every time it rains. Not recommended.
I second that, apply sparingly, and some patience, they will be dead in 3 days normally. It’s of course better to apply it before they mature.
The main advantage is that it has no herbicidal activity in the soil (of course that DOES NOT mean it’s not a problem in a water way, thus minimal amount).
Other than that, the most environmentally procedure (taking account raising acidity in the soil, etc) I am aware of is using a simple weed burner. You have to do it more often but you’re not dealing with often unexplored areas of what if I dump 5 gallons of vinegar within 20m of a stream over 2 years.
I use a much cheaper version of this that you can buy at home improvement stores (along with little propane canisters that fit it). Pluses include no pesticide exposure or chance of drift onto valuable plantings. The only real negative is that some deep-rooted weeds can grow back if only their tops are scorched, so you’d probably have to re-treat more often than with Roundup.
My suggestion would be to first kill the weeds with one of the above suggestions then or just use a weedeater to get rid of the bulk. Then use a pressure washer to both clean the dirt and grime from the paving stones and to dislodge the dirt/soil from between the stones. This is what is allowing your weeds to grow.
Be careful here so that you don’t blast too much out and loosen the stones. Then get some sand/stonedust and sweep it into the spaces.
Killing the weeds will be a short term fix that may not last the entire summer. Dormant seeds may start growing immediately and new seeds will be blown in quickly as well and you will be right back where you started.
CM
I don’t know how well it would work (or how invasive it might be), but you could try planting something like creeping thyme. Once established, it is claimed to choke out new weeds, and it smells good when you walk on it. There are lots of varieties.
I would be very afraid of using that where there is a fire danger. Like Colorado, near a forest or my back yard.
Not much risk of fire here this year. Some years there is, and during those years we apply common sense and don’t spray flames around.
How about putting grout between the cracks? Seems like a safe and permanent solution.
I like rjk’s idea of creeping thyme. There are several other little plants that serve that function.
The only problem with that is it will hold water. The degree thats a problem is very dependent on where you live. He might come back next year asking us about getting rid of moss lol.
Besides burning them, which I would insist is a more environmental friendly way of getting rid of them compared to any herbicide (of course no compared to just, gasp, pullling them) As indicated, after getting rid of them, or as a primary measure of uprooting them, to me it seems a good pressure wash, which cleans and even digs into the joints, followed by sweeping sand over the area would be a really good plan.
Heh. According to my (SK) uncles, you could gargle with Round-Up five times a day and it wouldn’t hurt you. Hell, it might even do you some good. 
In truth, you’re not supposed to lay into the weed with the torch until they’re scorched into nothing; all they need is momentary contact, enough to cook its vascular system and burst cell walls. The weed will probably not look more than slightly wilted after enough torch-time to ensure its demise. Rapid death will follow.
Still, the torch-makers recommending having a running hose sprayer available nearby.
Unless you’re pregnant. There are some new studies saying Roundup can cause birth defects. I’m not using it while I’m pregnant (of course, I’d generally jump at any excuse to get out of weeding).
Weed killer is effective. To reduce spread of toxins, spread with a sprayer and spray each weed individually. I had an aunt who would use roundup with a Q-Tip and touch each weed individually.
I would recommend pulling individually and planting grass or putting sand down in between the blocks or whatever. They’ve got sand out now that sets almost like concrete. Haven’t tried boiling water, but that sounds like a pretty great solution if it works.