An l-good sequencing of a Steiner triple system of order v, STS(v), is a permutation of the points of the system such that no l consecutive points in the permutation contains a block. It is known that every STS(v) with v > 3 has a 3-good sequencing. It is proved that every STS(v) with v >= 13 has a 4-good sequencing and every 3-chromatic STS(v) with v >= 15 has a 5-good sequencing. Computational results for Steiner triple systems of small order are also given.