How has cloud technologies altered your business continuity strategy?

Yes, significant reduction in data center DR cost, since we rely on the cloud provider for those systems in the cloud.

From a business person's perspective it hasn't changed at all. If the SLA with our internal customer (the Business) is recovery back to operations within 24 hours, then we design our Cloud contracts to cover that possibility.   Over the last 5 to 6 years, I think the outages we have experienced have actually been about the same.  We had some outages from AZURE being unavailable, and the impact was the same as an outage when we hosted internally.   I recommend you practice your Plans just like you did when you hosted internally.  There will still be outages from some source, and to keep the business running you need to practice and implement plans that keep the business functioning within the agreed to SLA.  (Service Level Agreement)