I come from a relatively small country population-vise, but not so small distance-vise - it's about 1700 km end-to-end on a great circle, although the part which is built out with a network of electrified rail is only about 700 km end-to-end. So the electrified train network spans (as in the shortest distance between the two furthest apart points) the about the same distance as Texas, measured east-west. Not all of this is heavily trafficked (although most places see much traffic than one train/day, both goods and people - when you build the infrastructure, it tends to get used), and it's a quite mountainous landscape so it's much harder to build tracks than on open plains with dessert climate. This network is of course linked to similar networks in neighboring countries (which again link up with the rest of the eurasian rail network), so quite a bit of goods get moved this way, also long distances. Trade unions and agreements for free movement of people is in place, so crossing the border is about as much hassle as crossing a state border in the US.
And no, I don't mean a third rail - this is what you generally use for subways, not trains where the standard is to lift the electric lines above ground, connecting to it with a pantograph http://en.wikipedia.org/wiki/P...
Further, you'll find tons of non-electric trains in very urban settings in the US - take the Caltrain for example, which is serving the Bay Area in CA - no electrification, level crossings in the middle of residential areas, and painfully slow - in a densely populated and quite wealthy area, where road transport is already at or above capacity during rush hours and no good alternatives for building more hi-capacity roads. Why is it so? Why not modernize at least these lines?
Anyway, the main point is that to avoid "burning tons of diesel to power the trains", one way to solve that is to power the trains electrically, which also has the bonus of making it easier to move other types of goods.