Screening existing tenants

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Curious what the rule of thumb is on screening existing tenants. I currently have no background, credit checks or employment verification for 4 different inherited tenants. 

We're slowly repositioning the property from a D to a B, tenants we're used to really low rents and are scheduled for a second rent increase next month, everyone is month-to-month and have been consistent with payment for the past year. 

I'm almost through @Brandon Turner 's book on Managing Rental Properties and realizing I'm setting myself up for future problems if the tenants can't meet the 3x income minimum rental requirement. Which, I suspect at least two of them won't. 

So, do I hand everyone an application to renew tenancy and cut to the chase or wait it out? Pardon the semi-rhetorical question but do you issue a notice to terminate tenancy to a paying tenant on the assumption you're going to face problems down the road and swallow the cost of rehabbing a unit or two? or in other words, does the peace of mind in placing prescreened tenants into market-rate units outweigh to cost of an rehab and termination of tenancy. 

Thanks, 

Michael