As I will hopefully be able to point out, Japanese Shôgi players have from early times on thought about the quality of their pieces. Thus it shouldn't be a surprise that the eldest known text that mentions Shôgi (This text is the Kirinshô ('Notes on the Kirin ( a mythical beast)'), written by Fujiwara no Yukinari (972--1027), reprinted e.g. in: Zoku gunshô ruijô ('Continued assorted writings'), vol. 31.2, Tokyo 1926, p190.) deals with the proper way to inscribe Shôgi pieces. The text makes it clear that from the early 11th century on the pieces were quite small (it is recommended to put them into a holder), that they were inscribed with ink, that the upper side which gives the unpromoted rank of piece is to be inscribed in regular chracters, and that the lower side which gives the promoted rank should be inscribed in flowing ('grass') script. The oldest extant pieces are dated 1058. These have been found in 1993 during a dig at the Kôfukuji in Nara, the then capital of Japan. (The find is presented in Nara Kôfukuji kyûkeidai ('On the old grounds of the Kôfukuji in Nara'), in: Mokkan kenkyû ('Studies in inscribed wood') Vol. 16 (1994), p26--30. --- It seems safe to conclude that these pieces really are shôgi pieces, as no other idea could possibly explain the design and inscriptions of the pieces found there.) They have the same shape as modern pieces (elongated pentagonal wedges). The material is Hinoki wood (Japanese cypress). --- We don't actually know for what Shôgi variant these pieces were used, but is it widely assumed that they belonged to one of the Heian-Shôgi games.
The first text that sort of describes Shôgi games is the Nichûreki ('From the two chûreki') from about 1230. (Nichûreki, esp. ch. 13, in: Shiseki shûran ('Collected mirrors of historical materials'), 43 vols, vol. 5, p250.---The two Chûreki are two small history works from the earlier 12th century.) The first game it describes is a small variant known as Heian-Shô-Shôgi ('Small Shôgi of the Heian period ( 794--1185)') with six different kinds of pieces (King, Gold General, Silver General, Knight, Lance, and Pawn), the second is the Heian-Dai-Shôgi ('Large Shôgi of the Heian period') with 13 different pieces (all of the above plus Copper General, Iron General, Side Mover, Wild Tiger, Flying Dragon, Free Chariot, and Go-Between). This Heian-Dai-Shôgi seems to have become extinct until ca. 1300, where another Dai-Shôgi ('Large Shôgi') appears.(We know that from the Futsû shôdôshû ('Collection of sermons for everyday use'), quoted in: Saeki Shin'ichi, ''Futsû shôdôshû'' no Shôgi kankei kiji ni tsuite -- Kamakura kôki no 'Shô-Shôgi' to 'Dai-Shôgi' ('On the passages regarding Shôgi in the Futsû shôdôshû -- the large and small Shôgi in the later Kamakura (1192--1333) period'), Yûgishi kenkyû ('Studies in the history of games') 5 (1993), p2--7.) This new Dai-Shôgi is the one played on 15 by 15 board with 130 pieces. From about 1350 on there is still another variant: Chû-Shôgi with presumably 92 pieces on a 12 by 12 board. More variants seem to have been developed until 1443; in this year the Shôgi rokushu no zushiki ('Diagrams and Explanations on six kinds of Shôgi') (Shôgi rokushu no zushiki, Tsurumine Shigenobu (ed.), in: Zatsugei sôsho ('Collected writings on different arts'), Tokyo 1915, vol. 1, p198-211.) was published. Once much disputed, it now seems to have been accepted as genuine.
This brings us back to the pieces: the preface to Shôgi rokushu no zushiki says the work had been copied by Minase Kanenari (1514--1602), the piece maker of earlier times. He has left behind a list of the Shôgi sets he made during the years 1590--1602. (This list, the Shôgi koma no nikki ('Diary of Shôgi pieces') has been published in Ei ('The bow-stand'), vol. 6 (1978), no. 3, p34--42, under the title Sanbyaku hachijû nen buri ni yô no moku wo mieta atarashii shiryô ('Material newly discovered after more than 380 years').) The list contains not only the type of Shôgi for which the sets were made, and the customer, but also the materials used.
93% of all 735 sets were made from Tsuge (boxwood), the remaining 7% were made from Byakudan (sandalwood), Kuwa (mulberry wood), Kusunoki (camphor wood), and Zôge (ivory). Most of the inscriptions were made in China ink, but a few of them were in urushi (Japanese lacquer). According to Minase's list, he (together with his son and adopted son) made 618 Shô-Shôgi sets and 106 Chû-Shôgi sets, in addition to which two Dai-Shôgi sets, two Dai-Dai-Shôgi sets, three Maka-Dai-dai-Shôgi sets, and four Tai-Shôgi sets were made. Of the Shô-Shôgi- and Chû-Shôgi sets from his hand a few have survived until today, but of course genuine Minase sets are priceless.
It seems that the materials Minase used are the 'standard' materials, with the possible exception of ivory, as nearly all known early pieces seem to have been made from indigenous Japanese wood.(These materials continued to be used for the next few centuries to come.) And lots of pieces have been found: in the largest single find of Shôgi pieces (at Asakura near Fukui) 174 of them were unearthed. Interestingly, these pieces seem to have intended for use in a Shô-Shôgi with 42 pieces: only pieces for the modern standard Shô-Shôgi plus one Drunk Elephant have been found. (Short notes on the find in Kubodera Kôichi, Nihon Shôgi shûsei ('Collection on Chess in Japan'), Tokyo 1995, p84.) Of course quite a few more pre-1600 pieces have been found, but as a whole they have no unusual features that deviate from what already has been said.