What about going with it? dig a small trench, line in with plastic and river rock, and make a small stream bed. You could make it level with the lawn for mowing, or plant it with marginal plants (much more fun).

It sounds like the swale through your yard was planned and you are probably not allowed to change the way it accepts water from upstream. However you can improve the way it moves the water. You could create a man-made creek bed like Wolfe15136 suggests. This would move the water more rapidly and, if you line it, not allow the ground to get so marshy. I suspect that even with a lining it would be kinda muddy, but you could place stones to line the edges of the creek and allow the kids to use them as stepping stones over the mud. Flattening the swale is probably going to make things worse by allowing the water to spread out more.

I might consider a French drain through the area to channel the water to the grate.

Alexa

Here is a link that might be useful: French Drain

I like the idea of lining it with rocks to create a dry stream bed. That could be a very handsome landscape feature. Just be sure to check with your municipality before doing anything. Drainage issues become legal issues very quickly.

The stepping stones are a cute idea, I could see kids having fun with that.

Have you considered a rain garden? You could create a 'saucer' with dug-out soil and plant thirsty plants to soak up the water. We learned a little about these in our garden club recently. Check out the link.

Here is a link that might be useful: Rain Gardens

A swell might not legally "be" your property in a community. You may need to check before looking further into improvements or the like.

The dry creekbed is a good suggestion. Another suggestion would be: if it is your property (or you can appeal to whatever community authority that it is lessening the use of your property), perhaps having the grate installed higher upstream, and piping the water underground to the location of the current grate would solve the problem.

In general you can change the form of the water conveyance through your property as long as you don't change the way it works at the upstream and downstream ends. That is, you could convert the grass to a man-made stream bed, or convert it too a rain garden, or install a drain beneath it, as long as you don't create a condition where water that used to leave the neighbor's yard starts to back up. Similarly, you can't change the location or the amount of water that leaves your yard. For example, you generally can't reroute the water across your yard and release it to the neighbor's yard at a different location that where it goes now unless the nighbor agrees. However, it could take a different path through your yard as long as it ends up in the same place.

We get various magazines at work where they want us to advertise. I just happened to see today that the June issue (yes June...not sure why it's out already but...) of Handy (Handyman?) Magazine has an article on making a dry river bed to help with flooding just like in your case. www.handymanclub.com/

The gist of the article was you dig a wide trench, about 8-12" deep. Lay down a pond liner and fill it with gravel. Then top it off with finer, more decorative gravel and large rocks like you'd expect to see in a dry riverbed.

They had one with a simple little stepping stone bridge across it and with plants and flowers alongside. Was actually a very nice effect.

Jim