In fact, the days of the week in Japanese do refer to the planets, and the correspondence is the same as in Western languages. In Japanese, every day name contains the element "you", which means "luminary, shining body", i.e planet. For example, "getsuyoubi" is the day (bi) of the planet (you) Moon (getsu).

A bit of history. The Chinese named the planets after the five elements: water for Mercury, metal for Venus, fire for Mars, wood for Jupiter, earth for Saturn. Then someone decided to name the days of the week after the planets, including the Sun and the Moon, and to use the same same day-to-planet correspondence as in the West. At the end of the XIXth century, the Japanese borrowed this convention from the Chinese. The Chinese themselves abandoned it and use a simple numerical system today.

In the West, the correspondence between planets and days of the week was probably invented by the Babylonians, so maybe the Chinese borrowed it from the Babylonians. An interesting fact is that, while most modern Western languages have partially lost this correspondence, it has remained intact in Japanese. The following table lists the name of the days in several languages. Names surrounded by parenthesis do not correspond to planets:

Planet    Latin          Japanese     English       French
                                                   
Sun       dies Solis     nichiyoubi   Sunday        (dimanche)
Moon      dies Lunae     getsuyoubi   Monday        lundi
Mars      dies Martis    kayoubi      (Tuesday)     mardi  
Mercury   dies Mercurii  suiyoubi     (Wednesday)   mercredi
Jupiter   dies Jovis     mokuyoubi    (Thursday)    jeudi
Venus     dies Veneris   kinyoubi     (Friday)      vendredi
Saturn    dies Saturni   doyoubi      Saturday      (samedi)

See http://www.cjvlang.com/Dow/dowjpn.html for a detailed (and very interesting) explanation of this question.