Smith, who was given two years’ probation, is entering a contract year after one of the best seasons of his career.
He will miss games against Houston, Denver and Green Bay before returning for the Chiefs’ game against Cincinnati in Week 4.
The 28-year-old Smith started all 16 games for Kansas City last season and was a major contributor for a defensive unit that finished second in the league in scoring defence and never allowed a quarterback to throw for 300 yards all season.
Under the NFL substance abuse policy, first-time offenders receive two-game suspensions. But there can be additional discipline depending on the circumstances, and Smith was pulled over by police last year after his car struck and broke a light pole.
The Chiefs used a first-round draft pick on Marcus Peters in the offseason and Smith???s suspension is likely to result in the rookie assuming a prominent role in the defence as soon as the season kicks off.