While these are mostly not an option for people just visiting Japan, there are a few ways to get cheaper shinkansen tickets. This answer probably belongs on expats, rather than travel; but, the first two options are mildly applicable to travelers.
1) Go to a discount shop. Several of these near Ueno, Shinjuku, and other hubs. They buy/sell tickets, but the discount is usually only 5-10%. If you luck out and the shop is stuck with a soon-to-expire ticket that meets your needs, you might be able to save around 20%.
2) Buy the JR "回数券", or ticket book. This has 6 tickets of the same route in a 3-month window, at a mild discount (again, only 5-10% I think). If you have three people, you can do a round trip with these six tickets. But, if you are doing a very long round-trip, the Japan Rail Pass will likely beat this deal.
You may be able to get by with no Japanese language for 1) and 2); but, for the next two options, you'll definitely need someone who can speak/read at a high level.
3) Yahoo Auctions. A slightly-less-expensive version of the discount shop, with a more limited supply, and longer lag time between purchase and ticket delivery.
4) The JR East online system (Eki Net) sells some shinkansen tickets for as much as 35% off the normal price. These are for trains with more stops at the smaller stations (and, sometimes they wait for the super-fast train to zip by). You need to register an account, and the page is only in Japanese. I think the other JR companies have a similar system, but I am only familiar with this one. To get the 35% ticket, you pretty much need to book it the morning it goes on sale (one month before the departure). Some smaller discounts are available up until 2 weeks before departure.