You have a functioning rain garden! You've chosen the site, dug the garden and planted it with region-appropriate, flood/drought-tolerant natives! Now what?
Well, there is a small amount of maintenance involved. As with any garden, in the first year when the plants are new you will probably have to water. No, the irony of watering a rain garden is not lost on me. But unless you get regular rains at least once a week or so for the entire spring, summer and fall, you will probably need to water at least a little bit to make sure the plants are established. Like any other garden, observe your plants to know when watering is needed. Are their leaves drooping? Do they look dried out? If so, it's time to water.
Once the garden is established, you can expect the normal type of maintenance you would with most perennials: weeding and dividing. I do my best to mulch my rain garden because it helps cut down on weeds, but with so much moisture in the soil you can bet that weeds will grow. Be prepared.
Also, if your rain garden plants are really happy they can end up looking like this:
out of control! They can get floppy, rambly, and generally unkempt looking if you let them (this is a reality of many native plants...they often do not have the prim habits of their more cultivated cousins).
If this happens you should be pleased that your plants are doing so well, and then you should do whatever you need to make your rain garden more aesthetically pleasing. That can mean dividing sedges, zizias, etc. until they're in smaller and more manageable clumps; it can mean removing overzealous plants like Physostegia; it can mean cutting back things that are getting out of hand. This year a couple really hard downpours knocked over a bunch of my zizias, so I just cut off the broken stems and flowers. The foliage still looks nice and there is no reason for undue messiness. The roots are still absorbing plenty of rain water.
However, you shouldn't be shocked or uncomfortable with a "naturalistic" look if you're growing a rain garden with native plants. It's going to happen. So keep it looking "pretty" naturalistic. For example, I like this:
Lastly, be aware that mulch floats. As I mentioned before, mulch helps cut down on weeds and I think it just makes gardens (rain or otherwise) look nicer. But yes it will float around when the garden is full of water, and after large rains you will need to clear it off plants and redistribute it. This really isn't difficult or time-consuming, just something to be aware of.
And that's it! Now you know how to create a beautiful, ecologically friendly rain garden! Please don't hesitate to leave further questions in the comments, and see the previous posts for more info:
Rain Gardens 101: How to Choose a Site
Rain Gardens 102: How to Dig the Garden
Rain Gardens 103: The Plants