EVA polymer is a soft, clear material that can be used for containers and tubing. It does not contain plasticisers but EVA bags can stick together or have their surfaced "roughened" unless a tough layer is added.
EVA bags cannot be steam sterilised.

Polyolefin mixtures

Medical Device & Diagnostic Industry

Better than PVC for storing blood platelets. but bags do not seal well enough for use with blood transfusions.
Plyolefin laminates have a sandwich of materials that allow them to be used for intravenous drip solutions. They cannot be steam-sterilised so are only used for heat-sensitive drugs made in sterile conditions.

Polyester

Polyester blood bags eliminate plasticiser but manufacturing difficulties and the fact that red blood cells survive longer in PVC bags mean it is not used for blood storage. Polyestes is an alternative to PVC in the delivery some drugs by intravenous drip.

Metallocene-based polyolefins

Similar properties to PVC but without the plasticisers. Being new relatively new materials, they will need to prove themselves before they are widely used. Concerns over safety, consistency and performance will need to be compared to PVC.