Be patient, keep your eyes open and keep your options open.
An REO can be a very good deal, but not always. Never get emotionally attached to a home or think that because you have offered full price that it is a done deal. Be prepared to wait for answers and then jump through burning hoops when the bank says "jump".
You just never know, and that is what can be so frustrating about dealing with bank-owned properties.
One word of caution: Be sure that offering over their ask is still market value. You've hopefully done your homework, but don't automatically assume you are getting a spectacular deal just because it's REO.
Good luck; let us know what happens!