Let me preface this by saying I'm not a Mountain Dew drinker. Really the only time that I can think of that I actually drank Mt. Dew and said "Wow, this is good!" Was after being in Italy for a year and a half drinking the bitterness they pass off as soft drinks.
I shouldn't say that, Italian beverages are fine. Some are an acquired taste, but there's nothing inherently wrong with them. They do run light on the sugar, though. So it's possible that my taste of Mt. Dew that made me think it was a good drink was based on sugar content. I love sugar. When I returned back to the States and acclimated myself to standard US syrupy goodness, Mt. Dew was just kind of *meh*. So this may not be an exact match, but it's a start for anyone looking for a match.
2 cups sugar, inverted in 1 cup water (with 1/4 tsp cream of tartar, see recipe below)
2 oranges (about 1/2 cup)
2 lemons (about 1/2 cup)
2 limes (about 1/4 cup)
1 cup water
Carbonated water up to 1 gallon.
Combine sugar, 1 cup water, and cream of tartar. Bring to a boil and simmer 20 minutes. Allow to cool. Bring the other 1 cup of water to a boil separately. Add the citrus peels, remove from heat and cover. Allow to steep until it has cooled to room temperature. Strain and add to sugar solution, along with the strained juices Ensure it is well blended. Add to carbonated water and mix gently until dissolved.
I'm thinking that this does need some work, but it's at least a start. I know some people online have been looking around for something like this, so here it is!
Update July 2014: Because this is such a popular recipe, I've included it in my book - Making Soda at Home. The recipe there takes on a slight variation and also includes methods for carbonating via fermentation or force carbonating like pre-mix rather than just making a syrup to add to carbonated water.