I'm a sucker for doing this kind of work once. If your swale starts to erode into a rushing creek or if it's "slow death" over a longer time, then you'll be into Plan B anyway.
Plan B for you is Plan A for me. Making the assumption that your lot has enough of a downhill grade from back to front to get the water out, then I'd dig a trench about 18" deep. Then I'd center-line it with a 2 foot wide piece filter fabric (you'll see what I mean in a moment.) Then I'd put in 3 to 4" of 3/4" gravel, then I'd run 4" flexible, perforated black pipe the whole length to the front street. Make sure the perforations are on the BOTTOM when you lay it in the ditch. Then I'd wrap the gravel and black pipe in the filter fabric (let it over-lap on top of the pipe. THEN, I'd back fill the remainder of the hole to whatever height you want to either 1) plant new grass, or 2) put back the sod you dug out. Do a little 'tamping' as you back fill the dirt every 4-6". Otherwise, you'll end up with a slight depression from settling. What you do when you get to the front of the house with the open end of the pipe is up to you. If you have a curb, well, you may have a small problem. If not, then just worry that you don't dump water on your neighbor's lot because in a lot of jurisdictions, you can be sued for "altering the natural flow of water". Etc., etc., etc..