They don't do relics the way that Catholocism does. Yeah, people may make a pilgrimage to somewhere that's assiciated with an important holy person. But I think the general Japanese cleanliness fetish put a kabosh on carrying around pieces of saints or bits of something they once wore. ('Cause, um, ick? Holy sweat is still sweat. And carrying around pieces of someone else's body is just gross. Even if it is contained in metal, like the crosspiece of a sword. Not to mention the cross symbology totally fails to work in Japan, since 1) not Christian, and 2) Japanese swords don't have a cross-shaped guard.)
They do have something called a "goshuuin", which is a special piece of calligraphy that will include the following information: The name of the temple or shrine; the date that the person visited said location; and some phrase or saying that is associated with the main deity venerated at the shrine or temple. The name of the temple may be contained within a stamp. Pretty much all goshuuin will have some stamps on them. I'm not quite sure what the fascination is, but the Japanese have a certain fondness for ink stamps. The goshuuin may have additional information in it as well, like the location of the temple, or other stuff that makes the temple unique or interesting.
For example, my goshuuin from Chuuzen-ji has a stamp on it reading "Bandou #18", since it is the 18th temple on the 33 temple Bandou pilgrimage. (Bandou is an old name for the area that is today essentially covered by the Kantou region.) It also has calligraphy reading "Mt. Nikkou", since it is located in the area called Nikkou mountain. (The tallest peak in the area is actually named Mt. Nantai. But the whole area is also generally refered to as "Mt. Nikkou", particularly in the goshuuin. In fact, all the goshuuin I have from the Nikkou region have "Mt. Nikkou" in them, except for the one from the Futaara Shrine, with leaves the "Mt." part off. But then, goshuuin from shrines are generally a good deal less elaborate than ones from Buddhist temples, and by that standard the one from Futaara Shrine is downright fancy.)
Goshuuin serve as a momento of having visited the temple. (Or, for ones that are on a pilgrimage, a receipt, as it were, proving that you did, in fact, visit said temple, along with when you did it.) They cost 300 yen each. Each one is unique, and, depending on the skill of the calligrapher, can be quite nice as a piece of religious artwork. Many temples -- and particularly ones on pilgrimages -- sell a book, called a goshuuin-cho, which contains a piece of paper which has been folded multiple times to create pages. Each goshuuin will fit on one page, and the entire book is around the size of your average paperback (albeit a really short one.) This way of putting the pages in does allow you to stretch the whole thing out and display all the goshuuin as one continuous scroll-type thing.
For the Shikoku pilgrimage, there is a special thingie that I know you can get. And my knowledge of said special thingie is so detailed that I can't even tell you what said thingie is called. It's done on something similar to the Japanese signboards (that you use for getting autographs), and tends to be a good deal more elaborate than a simple goshuuin, as well as more expensive. (I believe each one costs something like $10-15.) Once the pilgrimage has been completed, there is a special (and expensive) way to mount and frame all of them, and it turns into what is supposed to be a really nifty piece of artwork as well as a nice momento of one's journey. While it would be unbelievably cool to do this, I think I will probably skip doing so, since 1) I'm not going to be doing the pilgrimage in order, nor in only one trip, and 2) it's really quite expensive. I believe I've heard that the total cost of one of these, including mounting and framing, comes to about $6000. While it is art, I'm thinking that the whole pilgrimage will be quite expensive as is, so I will probably skip the fancy thing and just content myself with my goshuuin-chos. (Yes, I'm going to have to have at least two books for all 88 temples.) Just doing the goshuuin will probably cost around $300. And that's not counting temple entrance fees (which some of them will have), or the cost of getting to some of these places. 'Cause Shikoku is not one of the most transit-blessed parts of Japan. (In fact, I use examples of some of the places I've been to on Shikoku as examples of the most obscure places in Japan that I've been to.)
Was that more information than you were expecting?
Oh, yeah. Related to pilgrimages, it is not required to do the entire pilgrimage in one trip. While it's encouraged (and vastly cooler) to do the temples in order, it also isn't required to do so. Nor is it required to do the pilgrimage in the traditional way. (Read: all on foot.) Japan being Japan, there are actually tours that specialize in whisking pilgrims from one temple to the next in air-conditioned busses. What's important is the journey. Obviously, the more effort that you have to put into the journey, the more you will get out of it. Which make sense, if you think about it. You will do a whole lot more character building walking all around the countryside (and will experience a whole lot more of life) than you will whizzing around on a tour bus.