For what it's worth, I bought my first property back in April, and it came with existing tenants. It is true that you have to honor the lease, but in my case, the risk was reduced because it was tenant at will.
I may also be naive, but I like to give people a chance. And I'm glad I did, because they have turned out to be great residents.
Three things I did right away upon buying the property:
1) Meet with them. They have lived there longer than I have, so they were questioning me just as much as I was them. Meeting them puts a face to the name and forms a human connection... you're not just after there money.
2) Provided them with a rental application. I gave it to them so that I would know how to contact them and to get emergency contact information. Again, it's about establishing a connection.
3) We did a property walk-through. I had them show me all the problems they currently had with their apartment, then we identified the top 3, and got those fixed within a week. Again, you're building an image, it shows you care and your quick to act while allowing you to learn more about them.
Hope that helps!
Clinton