Funny you should ask, DK. I was going to actually take some pics of my weeds and ask you what they were (based on some past posts you've made, I thought you might know)
No problems at all. Where are you located - some are specific to certain areas? I found a site for California, but I've had terrible luck at finding anything that's an "all-in-one" picture identification chart for weeds online. I've got several charts at work, and yours may be something common.
The reason I say you may not want to use that tool is that some weeds spread by rhizomes and by cutting off the weed in one spot, you may cause TWO to grow elsewhere as it tries to survive and branches out underground, making it seem like you're gaining ground, but you're actually compounding the problem
If you've got a pic or two, I should be able to I.D. it, or check here to see if you can walk through it and identify it:
http://www.ipm.ucdavis.edu/TOOLS/TURF/PESTS/weedkey.htmlDown near the bottom there's a "begin key" box. Click on that, and it'll walk you through a bunch of steps to narrow it down, hopefully leaving you with one or two possibilities.
One thing - does the thing smell, either just standing near it, or when you tear/cut a piece of it off? I've never heard of a dog liking those types of weeds, but there's a first time for everything.....and if so, those are rather invasive weeds

I've used round up in the past, but now have a dog (who likes to eat the weeds) and I'm just a little leery of spraying the back yard.
Is there a website with photo examples of weeds? If so, I could point out which ones I've got.
See if you can find the local college's extension site for your area (the extension office is the agriculture/landscape-y help site - if you had soil samples you needed testing, that's where you'd send 'em) and they'll almost certainly have very specific information to your area about weeds and just about anything green and growing.
If your whole backyard is terrible and you want to kill everything off and reseed/sod and don't want pesticides for the dog to ingest, find a carpet store in your area and take a bunch of their remnants and lay them down over the area you want to kill off. It'll take a few weeks or so, but it's the safest way to kill off everything green underneath the carpet.