This was part of the reason why it took so long to discover the cause of Alexander Litvinenko's death by Polonium-210 poisoning. Doctors (and later detectives) had suspected some form of radiation poisoning, but the early tests used Geiger counters and came back negative. But Po-210 decays almost exclusively by emitting an alpha particle which is not detectable by Geiger counters. (And it also means it's not very dangerous outside the body but becomes extremely toxic if ingested.)