In the Qur'an, Islamic scripture commands the faithful to mark time by the cycles of the moon:

     They ask thee
     the New Moons
     Say: They are but signs
     To mark fixed periods of time
     In (the affairs of) men
     And for Pilgrimage.   (II:189)

     The number of months
     In the sight of Allah
     Is twelve (in a year)
     So ordained by Him
     The day He created
     The heavens and the earth;
     Of them four are sacred;
     That is the straight usage
     So wrong not yourselves
     Therein, and fight the Pagans. (IX: 36)

The Islamic lunar calendar has historical and spiritual significance. First, it is a reflection of the lack of agriculture in this nomadic society. The calendar is not tied to the seasons of the solar year, which agrarian societies depend on and which were the probable origin of the first human calendar systems. Since Islamic culture had no need to mark the seasons – and some wags may suggest that Arabia, where Islam originated, has only one season anyway – they could abandon without consequence the solar calendar used by the unbelievers.

Second, their calendar serves much the same purpose as the Christian season of Lent – to remind the faithful of sacrifice and to prepare them for sacrifice in their own lives. Early Muslim holy men, when fashioning the modern Islamic calendar, chose to start it on July 16, 622 C.E., the year of the HegiraMohammed’s flight from Mecca – rather than Mohammed’s birth, ascension, or revelation. This notion stands in stark contrast to the Christian calendar, which nominally begins in the birth year of Jesus Christ, and symbolizes rebirth and redemption for all believers.

Not only does the start of the calendar call Muslims to contemplate sacrifice, but so do the months therein. Of the Islamic months, four are sacred. To Westerners, the best-known of these is the month of Ramadan, during which believers fast from dawn to dusk and focus on worship and contemplation.

Third, the Islamic calendar enables believers to comply with the will of Allah as written in the Qur’an. The scripture sets out strict instructions on how Muslims are to live their lives during the sacred months. The only way for Muslims to know when these instructions apply is to carefully track the lunar months. If they fail to do this, they are like the unbelievers:

     Verily the transposing 
     (Of a prohibited month)
     Is an addition to Unbelief:
     The Unbelievers are led
     To wrong thereby: for they make
     it lawful one year,
     And forbidden another year,
     Of months forbidden by Allah
     And make such forbidden ones
     Lawful.  The evil of their course
     Seems pleasing to them.
     But Allah guideth not
     Those who reject Faith.  (IX: 37)

Without knowing, for example, when Ramadan falls, no man can obey the Qur’an’s call to fast during that month. Such a person might fast at the wrong time and fail to keep fast at the right time, failing to submit to Allah’s will. In this way, unbelievers add wrong practice to wrong belief, says the Qur’an.

Thus, the keeping of time by the lunar cycle is both critically important and distinctive to Islamic faith and culture, and provides an important reminder of the will of Allah.

Sources:
http://www.cob.ohio-state.edu/~muhanna_1/hijri-intro.html
http://www.encyclopedia.com/printablenew/05768.html
http://www.holidays.net/ramadan/story.htm