This proposition, if taken too far, is an instance of the doctrine of antinomianism (see Webster 1913 on Antinomian). How someone arrives at the antinomian position depends upon their theological tradition, but it appears in many different denominations and sects.
For example, in dispensational circles it is argued that the law and the new covenant are distinct and unrelated dispensations of God's grace, and therefore the old covenant law is not binding upon partakers of the new covenant.
If you have more of a covenantal theology, then you must be very careful before flatly dismissing adherence to the law, because the new covenant builds upon (instead of replaces) the old. Those who would say at least part of the OT law is somehow meaningful binding upon mankind hold to a doctrine of theonomy (and thus are theonomists), the most (I believe) compelling form of which is theonomic reconstructionism (a doctrine appearing frequently in reformed theology). To a reconstructionist, the law is separated into ethical and ceremonial/illustrative components (as noted by dann); the former remains binding as principles for civil conduct and ethical life, the latter was binding only upon pre-messiah Israel. Thus, while a Christian is free (as argued very persuasively by dann above) to walk afoul of the entirety of the Mosaic law without consequences as to his eternal salvation, he may be positioning himself in opposition to the civil judgement of God1, which could lead to his experience of this present life being rather badly fouled up.
1 Another way to think of this is through the lens of what are called spiritual laws; this is the notion that God's ethical laws have material consequences, in the form of biases or "drifts" in the natural situational order. For example, tit for tat is a spiritual law of rabbinic origin which basically says "what goes around comes around" or (in more Christian terms) "God is not mocked - as a man sows, so shall he reap."