Scripture says nothing about a required mode of burial for either believers or non-believers. However, burying the body was the standard practice among the Israelites in the Old Testament and Christians in the New. There were some exceptions: the people decided to cremate Saul and Jonathan and then bury their ashes because their bodies had been mutilated by the Philistines (1 Sam. 31:8-13). In another instance, Achan and his family were cremated after being executed for sinning against Israel (Josh. 7:25).
Obviously any buried body will eventually decompose (Eccles. 12:7). So cremation isn't a strange or wrong practice-it merely accelerates the natural process of oxidation. The believer will one day receive a new body (1 Cor. 15:42-49; 1 Thess. 4:13-18; Job 19:25-26), thus the state of what remains of the old body is unimportant.
The imagery of Christ's resurrection pictures burial and then a raising up from the dead (Rom. 6:3-5; 1 Cor. 15:3-4). Because of that, many Christians prefer burial to cremation to maintain a likeness to Christ's burial (although literally He was laid in state in a cave, not buried in the ground).
What we need to focus on as Christians is not how to dispose of our earthly bodies, but that one day new bodies will be fashioned for us like our Lord's glorious resurrection body (see Phil. 3:21; cf. Luke 24:30-40; John 20:19, 26; 21:1-14; and Acts 1:1-9 to get an idea of what to look forward to). That transformation will be eternal!