I am a clinical-year student. I have tattoos from my shoulders, down to each of my elbows and one on my forearm. Very visible.
Experiences my vary:
I'm in the south. During didactic I covered them for the first week, then never again. In clinicals I wear my white coat all the time because we're required to do so. On the times when I haven't worn my white coat, surgery/emergency medicine, no one has cared or said anything to me. I've had at least two preceptors who did not care in the least. I do carry a white long-sleeved under amour shirt in my bag at all times in case it's needed. I have seen nurse wearing long sleeves to cover tattoos on my rotations, but I have not seen providers with exposed tattoos either. Most times, you'll be wearing a dress shirt, +/- white coat, so not a biggie. Also, when you interview you'll be wearing a suit and unless you divulge your tattoos, they won't know - personally I wouldn't submit that information.
tl;dr - haven't had a problem yet.