There is no standard term. There is no a strict association between "x axis" and "row". (For that matter, rows are generally indexed vertically, which would associate them with the y axis, not the x axis.) A figure (or feature space) with a y axis won't necessarily have anything you could point to as a "row".
Pixel-based images do have rows and columns that are often indexed with y,x values, in which case z often means depth (thus, "layers"), but that's specific to graphics.
If you're talking about matrices, your question was asked on reddit a while back, where the favored (but non-standard) term was "aisle":
As others have noted, Excel uses "sheets" for its third dimension, following the paper metaphor.
When working with three-dimensional data sets, the third dimension is defined by the type of data. For example, hyperspectral imagers create data cubes that are indexed by row (line), column, and band (wavelength or channel). A scientist could say
"Add another band to the data set for observations at 2.4 microns."
But that's specific to that data type, of course. Perhaps your best strategy would be to identify the appropriate specific type of dimension for your needs.