No such thing as a "simple refi" (unless streamline VA, FHA, or Home Affordable Refi Program HARP), any refinance is the same process as a purchase (underwriting, appraisal, etc). With the scenario you lay out, it's rather simple. If you recently started renting out the property and the rental income is not listed on your tax returns, then you'd have to prove that you have been receiving the rent from the new tenant in order to use the payment to offset the debt for qualification purposes. If it doesn't matter (ie, you have enough income to offset all your debt including your new housing payment), then you're good to go. If you need the rental income to offset your debt ratio and don't have the income on your taxes or can't show the new rental income received, the property has to have 70-80% equity (depending on the loan type you go with).
I know I've said a lot, but again, it's a rather seemless process. Expect it to take 30-45 days to close.