It becomes part of your routine pretty quickly. Also, our Techs can draw labs. Triponins in our facility are every 8 hours, so conceiveably, you would have 1 extra blood draw per shift. There are lots of ways to incorporate this into your daily routine--we also do our own labs, our own IV's, etc. If you are a night shifter, you have one triponin to draw, and might as well draw for the am labs as well if your policy says you can do this. If you are rotating IV sites, draw off that, again if your policy allows this.
Perhaps if you have one aide on the floor per shift, they can do the draws for the floor if your facility policy allows them to do so?
I would feel badly that your phlebotomist's are losing their jobs. However, for a number of nurses drawing labs is part of the job that we do. The Joint Commission has some rules regarding this--mostly revolving around patient ID stickers need to be at the bedside, and labs labelled accordingly at bedside.
Otherwise, I would ask what the new policy is, get a copy of it, and just be sure--much like giving meds, that they are properly labelled and sent, entered in computer--and again, it becomes part of the daily routine when you assess your patients.