Solution 1: Buried pipe
I had this exact problem and had to remedy it with buried drainage pipe. I put in a surface drain at the lowest spot in the back yard that catches most of the surface runoff, and then hooked up about 30' of perforated corex pipe from the surface drain box to other swampy areas that were slightly higher. From the surface drain I ran about 110 feet of solid 4" corex (buried 1-3' underground to keep it graded persistently downward) to my front yard where there's a convenient pipe under the sidewalk that drains down the street to a city storm drain.
You need to ensure that every length of pipe keeps up at least a slight downward grade so that you're not just putting pipes in the ground to fill up with water. I've seen 1" drop per 8 feet of run recommended.
This solution works really well. Some people have concerns about the longevity of corex pipe due to difficulty cleaning it with bladed pipe cleaners, but after 3 years mine still works 100% and does not appear to be clogging with sediment anywhere.
Solution 2: Dry well
If your volume of water is relatively low, you might be able to get away with installing a dry well, which is basically just a big cistern underground that can capture water and slowly leech it into the ground. If your clay is deep and you can't dig a dry well deep enough to drain into something other than clay, it won't drain fast enough and it will fill up and become worse than useless. If your clay is really just 5 feet deep, this should work fine and be less work than running huge lengths of pipe - you'd just dig a big hole to install the dry well and run a few pipes to drain into it.