Sri Lanka were due to begin their first innings after bowling out Bangladesh for 131 before lunch. Off-spinner Muttiah Muralitharan claimed 6-28 and left-arm seamer Sujeewa De Silva took 2-29.
The poor drainage and coverage system at the Asgiriya stadium left the grounds staff struggling to dry the muddy outfield in the brief spells when the rain stopped.
Umpires Rudi Koertzen and Suresh Shastri decided not to fix any inspection till the weather cleared.
Sri Lanka enjoy a 2-0 lead in the series after winning the first two Tests by innings margins.