The biggest issue with ionizing radiation for living creatures, is damage to DNA. Any creature with a sufficiently redundant DNA sequence, and extremely active repair functions would be able to correct and heal any damage before any real issues.
See the wiki on radiation poisoning for more information.
@anon made a point that warrants an edit to add a link. There are two categories of radiation that must be shielded differently. Your approach must target the type of radiation you're worried about. radiation shielding overview