A plurality of sealed blood sample tubes are housed in a rack and a plurality of these racks are vertically stacked, with the sample tubes lying horizontally. The racks are successively deposited onto a horizontal conveyor belt which is housed in and moves longitudinally on a table that rocks around its longitudinal axis to mix the samples in a semi-inverting mode as a rack is stepped from the stack to a sample aspiration station and/or thereat. The aspiration station preferably includes a sample segmenting and diluting valve and a sample identification reader. Aspiration is accomplished by pushing a sample tube partially out from the carrier rack and onto a seal piercing tip of the aspiration probe. The tube then is returned to the rack. The other tubes in that rack similarly are aspirated after the rack is stepped to align each tube with the aspiration probe, with table rocking and sampling mixing accompanying each advancing step of the rack. After a rack has advanced through the aspiration station, it is stacked out of the way.