This is a photo of the lawn at the National Mall in Washington and it's organic. There's no reason why you can't have a great organic lawn, it just means using some different products and cultural practices. The key to a great lawn without chemicals is fertility and pH. You've got to give the grass an environment that will let the plant thrive and choke out the weeds. When grass has everything it needs, it will outgrow anything. The broadleaf herbicide 2,4-D is the most common chemical weed killer used on lawns today and there's lots of research telling us that it can be harmful to us. It's something I always tell gardeners to stay away from.
The first step to converting to a chemical free lawn this time of the year is to...
...do a core aeration on the turf. The aeration machine can be rented, I usually get together with my neighbor to pull cigar shaped plugs out of the lawn. This helps ease compaction and allows water and fertilizer to get to the roots. The next step is to add a good organic fertilizer. They are easy to find at a good nursery and there are four-step organic programs available. Overseeding is another technique that will help the grass overtake the weeds. Spread grass seed over the lawn and rake it in, the new sprouts will grow fast and fill in areas that are bare. Check your soil pH, there are pH meters available at local nurseries and hardware stores. When the pH is off, that's when weeds and moss can thrive, change the pH to neutral and the grass will thrive. There's a great weed control if you're not overseeding called corn gluten meal. It's a byproduct of the corn milling process and the granular product stops weeds from sprouting. That means all seeds, not just weed seeds. It's applied in the fall and spring stopping annual weeds from germinating.
During the growing season, keep the grass high, three and a half inches if possible and then cut it a little shorter as we get to the end of the season. The long grass will shade out the weeds. These are just a few tips, there's lots of information here. Give an organic lawn a try, it's a better way to enjoy soft green grass.