The Schengen rules have certain exemptions for citizens of countries which concluded bilateral visa-free agreements before the Schengen area took effect. For example, a relevant quote from the Czech Ministry of the Interior website:
In practice, it means that, for instance, a citizen of the Korean Republic can travel in the Schengen area (including the CR) for a period of 90 days in any 180-day period. If the citizen of the Korean Republic stayed in the Schengen area for the entire period, he/she can move to the CR before the 3 months are up and then stay for a further 90 days in the CR without a visa. During these 90 days in the CR, he/she can no longer travel to other Schengen states without a visa. In the event of departing and returning to the CR during this period, it is necessary to take a direct flight.
The provision above applies to citizens of:
Argentina, Chile, Croatia, Israel, Korea, Costa Rica, Malaysia, Uruguay
What other countries have similar provisions and to citizens of which countries do they apply? And does the Schengen clock "count back" while the tourist is using his rights as part of a bilateral agreement?
Note that since the rule still precludes staying for more than 90 days in any single country, this is on topic for Travel.SE.