There's a much faster solution to this than any other below.
In the "Par for the Course" map, go to a anti-air turret and spread fire at the trees. You can destroy 100 trees in 30 seconds by doing this.
Trees ran over with vehicles apparently don't count towards this achievement.
Check your progress at:http://badcompany.ea.com/myawards/xbox/YOUR_GAMERSCORE_HERE/...