Plat maps won't help. Nor will most of the info available publicly online. I've worked for a civil engineer for 10 years, and the bottom line is that the information is so spread out, and in some cases nonexistant, that you would never be able to do this.
The first problem is that in many cases the easements are so old, and the deeds so difficult to read, you could spend hours piecing together the right of way over one parcel of land. Many times there are multiple easements as they were added-on over the years. They are also shared by different utilities.
So for example you'd find one document that grants Verizon the use of a railroad's right of way. Then you have to pull all of the deeds for the railroad (hundreds or thousand per county) and try to put them together. Those old railroad deeds will say something like "the east 99 feet of Farmer Smith's property, in so-and-so a section." Then you have to go pull Smith's old deed, which says "40 acres, lying south of the river, and east of Farmer Johnson's land, and north of some other guy's property." And no, those are not exaggerations at all. In short, you'll be putting together the puzzle pieces for weeks, and then you might have a single line along a railroad done for your county.
Add on top of that, many street right-of-ways are just assumed. Sure, maybe there are some old deeds that grant the right-of-way for each road over each parcel of land, but again, you'll be putting together a giant puzzle with pieces that don't fit together well.
In short, good luck. You'd be better off just taking a map and hi-lighting all of the roads, assuming that at least some communication lines follow each road.