I have paver bricks with this cracks between each brick and weeds.
Handling the weeds is a challenge for my wife and I.
Our strategy now is:
hand pick the weeds
use a power washer
fill the cracks in with locking sand
This is the rare place where a weed torch/flame weeder actually makes sense - though plain old hot water will work as well - and neither will attack your pavers the way that acids will (unless you go nuts with the weed torch - so don't go nuts with it - weeds do not need to be burnt to a crisp to kill them.)
If you happen to own a steamer (that puts it out via a hose or wand) of some sort (old wallpaper tool, or infomercial impulse buy miracle cleaner - I can't really see applying the espresso-maker's steam wand to the patio) that would be ideal (really hot water...)
Try horticultural vinegar. It is a much stronger acid than regular store bought vinegar. You can buy it online. Spray the weeds liberally in the middle of a hot sunny day. Hot direct sun is the most important thing when using horticultural vinegar. I'm not sure why stormy is suggesting glyphosphate herbicide when you said you don't want to use bad chemicals. Glyphosphate to me is the epitome of "bad chemicals".
You could also fill the patio cracks with plants that you do want there, which will out compete the weeds for the most part. You can get sedums and break their roots apart and smash them in between the pavers. They will grow in sand just fine and will spread to fill in the cracks.
Another option is to fill those cracks with a polymer sand which is solid and won't allow organic material to accumulate for plants to grow out of.
I have not had problems with my pavers because they were installed well with almost no gaps (one of the few things a previous owner did right), however white vinegar works as well as other chemicals in other places in my yard (cracks in concrete). Use an old spray bottle to apply it.
Since weeds in between pavers do not come from below, but from tiny seeds blown in, you might try mixing something like Preen in with the sand if the vinegar does not work well. (assuming Preen is not on your list of bad chemicals)
If you've got pavers, that suggests block paving (also known originally as paviours), the brick shaped paving often used on driveways and patios, rather than paving slabs - these are never pointed in between. They're closely laid, but with enough room for sand to be forced down the gaps by using a wacker plate machine. Over time, inevitably, the sand can disappear, and weed seeds will germinate.
Strip out all the weeds in whatever way you choose, rake the joints out and use a Paving Sand with Weedkiller, preferably shaking it in place with a wacker plate, if not be prepared for lots of brushing and brushing and brushing. That should keep the weeds away until the sand has become eroded again, and is less damaging than regular spraying or liquid weedkiller applications twice a year. This link is to a British site, but there will be equivalent weed prevention sands available where you are http://www.paving.org/index.php/paving-sand-with-weed-killer/
Be careful when using strong vinegars - they can cause a lot of efflorescence or permanent damage to pavers.
[These pavers have spaces between individual pavers? One thing you might do is redo the entire patio floor so there are no spaces. Need to send a picture of paver type and paver bed; how deep is the sand beneath, the gravel beneath that?
This will be almost the same amount of work that you are considering pulling the weeds. If there is no proper bed I seriously would pick the pavers up and make the correct bed. You'll need to rent a compactor, dig out the gobbeldygook, install 4" of 5/8 minus gravel, 2" of mason sand on top, compact well in layers. Then you begin to reinstall the pavers butted up against one another, no space at all. You'll need string and a 2X4 and one of those big soft hand mallets to tap an entire row tightly to the last row. Keeping a string all around the perimeter that you have leveled as the top of the patio. Using that string and the 2X4 and tapping complete your patio. The plastic edgers are nailed into the ground and hold the pavers in place. Big 8" long nails. Spikes I guess more appropriately. Your building materials place will have everything and people to help you.
By butting the pavers up together REDUCES the chance of weeds but will not stop weeds. Using the pressure washer one a year will. I am hoping these pavers are 2" thick and untreated concrete...
If your pavers are thinner then that sandcrete would be a good idea. 2" thick CMU pavers look terrible done with spacing.
The most professional paver look is without any spacing. If you are going to do so much work anyway it would be almost easier to just redo the patio and make sure the job is done right. A GOOD paver patio/driveway ADDS big value to your home! Please send a picture of what you are dealing with first, how large and what is the condition of the bed?
This is a low-effort solution I've used on some areas successfully (in southern England), and have been meaning to try on larger areas after being disappointed with weedkiller and unwilling to use it at the moment. I mainly use a wire brush with a long handle and/or a shaped scraper blade; you'd need to use those to finish the job but dead weeds should be weaker and com up more easily.
Cover it with plastic sheet (black or clear -- this link for the discussion) and wait a few weeks. It's best done in the height of summer to get the heat. My idea is to put the sheet at the weekend before going on holiday, and to take it up at the weekend when we're back. For a 2 week holiday that's 3--4 weeks, and as we all know, it's always sunny in the garden at home when you've gone abroad. This would avoid having it around too much for walking on and looking unsightly.
This could be a partner to other methods: I've got island beds with wooden edges so would have to leave a decent margin round them with any fire-based approach -- the plastic sheeting could go down in these margins after burning the larger areas (then it could be left longer, too). Similarly I wouldn't fancy using weedkillers that close to the beds even if I used them on the main patio.
Once they're laid, it's too late. But when the paving is being laid, ask the fitters to put a thin layer of sand/cement mix on the top of the sharp sand underneath. It doesn't completely stop weeds between the pavers, but it greatly reduces their ability to send down roots and grow bigger.